The linear spectrum of a quantum dot coupled to a nano-cavity G. Tarel and V. Savona Institut de Th´eorie des Phenom`enes Physiques, Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne EPFL, Switzerland (Dated: January 8, 2010) Wedevelop atheoretical formalism tomodel thelinear spectrumof a quantumdot embedded in a high quality cavity,in presence of an arbitrary mechanism modifying the homogeneous spectrum of the quantum dot. Within the simple assumption of lorentzian broadening, we show how the known predictions of cavity quantum electrodynamics are recovered. We then apply our model to the case where the quantum dot interacts with an acoustic-phonon reservoir, producing phonon 0 sidebands in the response of the bare dot. In this case, we show that the sidebands can sustain 1 thespectralresponseofthecavity-likepeakevenatmoderatedot-cavitydetuning,thussupporting 0 recent experimental findings. 2 n PACSnumbers: 78.67.Hc,42.50.Pq,78.55.-m,78.20.Bh a J 8 I. INTRODUCTION excitation density. Using a Maxwell formalism, we show howthiseffecttranslatesintheoverallspectralsignature ] oftheQD-cavitysystem,atvaryingdetuningandQDpa- r Thedescriptionofsemiconductorquantumdots(QDs) e rameters. Our model accounts for the specific shape of as isolated atomic-like quantum systems is largely over- h the resonant cavity mode as well as for the microscopic t simplified. The confined electrons and holes in a QD, in o parameters of the QD excitonic transition. fact, interact rather efficiently with both the electronic . t and vibrational [1–5] degrees of freedom of the semicon- We first develop the general formalism, assuming an a m ductor environment, in ways that can only be described arbitrary energy-dependent self-energy for the QD exci- beyond the simple perturbation theory. In addition to tonic transition. Then we apply the model to two cases. - d the semiconductor medium, QDs interact with the sur- The first is that of a lorentz-shaped QD line, character- n rounding electromagnetic field, especially if embedded izedbyaconstantbroadeningγ0. Inthiscase,werecover o in a photonic structure with sharp electromagnetic reso- the resultthat is well knownfromCQED[23]. Then, we c nances. As anexample,the electromagneticfieldcanve- assume a QD spectrum arising from the coupling to lon- [ hiculate an excitation transfer between two distant QDs gitudinal acoustic phonons, with the corresponding self- 3 withnon-overlappingelectronicstates[6–8]. IftheQDis energymodeledwithinasecond-Bornapproximation[3]. v embeddedinahigh-qualitycavity,the3Dconfinementof ThecouplingproducesphononsidebandsinthebareQD 3 electromagneticfieldcanleadtoobservationofthestrong spectralresponse. Weshowhowthespectralsignatureof 4 coupling between one QD and the resonant mode of the thesesidebandsisenhancedbythepresenceofthecavity 6 electromagnetic field [9–13]. This system, however, can resonance, even for moderate QD-cavity detuning. The 3 . not be seen as a perfect parallel to the atom-cavity cou- persistence of light emission at the cavity-like peak has 2 pling in cavity quantum electrodynamics (CQED) [14]. beentheobjectofseveralexperimentalinvestigationsre- 0 The semiconductor environment, in particular, can af- cently [13, 20–22, 24]. At small detuning, the acoustic 8 fect the system in several ways that have no analogous phononmechanismisexpectedtocontributesignificantly 0 : in its atom-cavity counterpart. A first effect is brought to this effect, as the energy width ofthe sidebands is de- v by the coupling of the QD to an external reservoir (e.g. termined by the exciton spatial confinement [15], and i X phonons)thatcanproduceasignificantchangeintheho- amounts to 1-2 meV in typical samples. This effect has mogeneousspectralsignatureofthedot. Asexampleswe been recently addressedusing phenomenologicaldephas- r a quotethebroadsidebandsoriginatingbythe couplingto ing to account for the modified QD spectral signature longitudinal acoustic phonons beyond perturbation the- [25–27], or within a more microscopic approach to the ory [2, 3, 5, 15–17], or the similar effect due to opti- phononsidebandmechanismatzeroQD-cavitydetuning cal phonons [18, 19]. In addition to these homogeneous [17]. The importance of our work lies in the fact that a modifications ofthe bareQD spectrum,other significant generalhomogeneousmechanismactingontheQDspec- spectral changes can arise when the QD is multiply ex- trum is modeled, and an explicit expressionfor the total cited,duetotransitionsbetweencontinuumstatesinthe emission spectrum, accounting for spatial and spectral wettinglayerabovetheQDconfiningbarrier. Thiseffect cavity form factors, is derived. occursalreadyatmoderateexcitationandleadstoasize- InSectionII,wepresentthegeneraltheoreticalformal- ableenhancementofoff-resonancelightemission[20–22]. ism. Section III is devoted to deriving the simple CQED Here,weaddressthefirstclassofthesesemiconductor- result in the limit of lorentz QD broadening. In Section relatedeffects,wherea homogeneouschangeinthe spec- IV, we study the case of a QD coupled to a reservoir of trumofthesingleQDispresent. Werestricttothelinear longitudinal acoustic phonons, and discuss how phonon spectralresponseofthecavity-QDsystem,holdingatlow sidebands enhance the cavity-like emission spectrum at 2 finite cavity QD detuning. In Section V, we present our B. Photon Green’s function conclusions. We introduce the in-plane Green’s tensor of the pho- ton (r,r′,ω), which is defined as the Green’s tensor of G II. THEORETICAL FORMALISM the Maxwell equation. We have previously shown that simple analyticalexpressionshold in the case of a QD in ahomogeneousmedium[6]orinaplanarmicrocavity[7]. A. Maxwell equations Inthe generalcase,acompactanalyticalexpressioncan- not be found. Formally, the Green’s function is defined WeconsiderthesystemofoneQDembeddedinareso- as : nantnanocavity. The cavitycanbe ofanykind(e.g. pil- lar [10], photonic crystaldefect[11], microdiscs[12], etc.), ω2 ′ ′ Υ(r) (r,r ,ω)=δ(r r ) (4) with the only assumptionthatone welldistinct resonant (cid:20)c2 − (cid:21)G − mode exists in the vicinity of the QD transition wave- length. Our objective is to derive the physical parame- where Υ(r) is a time independent, hermitian, linear dif- ters characterizingan effective coupling to this mode, at ferential operator that possesses a complete set of eigen- frequencyωc,fromthe microscopicdetailsofthe electro- functions {Φu(r)} where u is a continuous index. The magnetic field in the structure. We assume a QD lying set is considered as orthonormal. This differential prob- at position r0. Typically, this position is selected to lie lem belongs to the class described by Fredholm theory wheretheelectricfieldhasmaximumamplitude,inorder [30]. Itthereforeadmitsaformalsolutionintermsofthe to maximize QD-cavitycoupling. In the limit of low QD resolvent representation excitation, the spectra are determined by the linear op- Φ (r)Φ∗(r′) tical response, and are described by Maxwell equations (r,r′,ω)= du u u . (5) for the electric field coupled to the linear susceptibil- G Z ωu2 ω2 c2 − c2 ity tensor of the QD. Under this assumption, the steps leading to a set of coupled mode equations are formally OnceobtainedtheGreen’sfunctionofthephotonicstruc- the same as in our previous works [6, 7]. In particular ture, the solution of Eq. (3), corresponding to an input Maxwell equations are cast into an integral Dyson equa- field 0(r,ω) can be written as follows [28]: Q tion [28]. We denote with ǫ = ǫ(r) the spatially depen- dent dielectric constant that characterizes the resonant (r,ω)= 0(r,ω) (6) Q Q photonic structure. In the frequency domain, we have (assuming non magnetic medium and no free charges) : ω2 χˆ (r′,r′′,ω) +4π dr′dr′′ (r,r′,ω) QD (r′′,ω), c2 Z Z G ǫ(r′) Q ω2 p ∇∧∇∧E(r,ω)− c2(cid:20)ǫ(r)E(r,ω) Thekeyassumptionofourprocedureisthatonestrongly resonant mode exists and is energetically well distinct +4π dr′χˆ (r,r′,ω) (r′,ω) =0, from any other spectral feature (discrete or continuous) QD Z ·E (cid:21) of the structure under investigation. This is the case for allkindsofhigh-qualitynanocavities. Closetoresonance where (r,ω) is the electric field, r is the 3-D position ω ω , the following approximation then holds vector,Eand χˆ the 3 3 linear optical susceptibility ≈ c QD × tensoroftheQDsubsystem. Inordertodefineahermitic ′ Φ0(r)Φ∗0(r′)c2 problem, we adopt the standard replacement [29] G(r,r ,ω)≈ 2ω (ω ω iκ) c c 2 − − Φ (r)Φ∗(r′) (r,ω)= ǫ(r) (r,ω), (1) + du u u . (7) Q p E Z 2ωu(ωu−ω−iκ2u) and A similar expression was used by Sakoda et al. [31] and 1 1 Hughes et al. [32] . Here, we neglect the longitudinal Υ= . (2) optical modes, consistently with the exciton optical se- ǫ(r)∇∧{∇∧ ǫ(r)} lection rules that we assume (see below). In compact p p form we obtain This leads to the following hermitic problem: ′ Φ0(r)Φ∗0(r′)c2 ′ (r,r ,ω) +g (r,r ). (8) ω2 G ≈ 2ω (ω ω iκ) c Υ (r,ω) (r,ω)= c c− − 2 Q − c2Q 4πω2 (r′,ω) Theresonantcavitymodearisesassharpresonanceinthe ′ ′ + dr χˆ (r,r ,ω)Q . (3) energy-dependent density of the eigenmodes. We have c2 ǫ(r)Z QD ǫ(r′) characterized this resonance by a damping constant κ, p p 3 that models the finite lifetime of the mode. This step is (9) result in necessary, as we are approximating an everywhere con- tinuous mode spectrum with one discrete mode plus a (r,ω)= 0(r,ω)+ Q Q nonresonant continuum. Formally, this passage can be Φ0(r)Φ∗0(r0)c2 justified in terms of the quasi-mode theory [33, 34], by M(ω)(cid:20)2ω (ω ω iκ) +gc(r0,r0)(cid:21)Q(r0,ω) , assumingweakcouplingbetweenanidealundampedcav- c c− − 2 (11) ity mode and the vacuum electromagnetic field outside the cavity. In Eq. (8), g (r,r′) represents the contri- c with bution of all other modes, and is supposed to be small at ω ≈ ωc. A complete numerical calculation of cavity (ω)= 4πµ2cv ω2 , (12) eigenmodes, like e.g. that carried out in Ref. (35), can M ~c2 (ω0 ω iγ20)√ǫM be usedto testthis assumption. Inthe following,wewill − − expressgc(r,r′)asthesumofitsrealandimaginaryparts where ǫM = ǫ(r0). This expression is the starting point for a(r,r′) and ib(r,r′). As shown later, these are responsi- computing the spectral properties of the cavity-QD system. ble – respectively – of a shift and a broadening of the It gives direct access to the linear response spectrum of the QD emission spectrum. More precisely, the term b(r,r′) system. From this, the emission spectrum can also be mod- is responsible of the decay of the excited QD into the eled. continuum of background electromagnetic modes. This determinesthefreedecayrateoftheQD,usuallydenoted B. Emission spectrum as γ in CQED. We first take all fields at position r . Then, Eq. (11) can 0 berewritten in compact form III. CQED LIMIT γ κ Q(r0,ω)((ω˜0−ω−i2)(ωc−ω−i2)−g2) (13) As a simple test of our formalism, we can recover the γ κ limitofonetwo-levelemitterinaresonantcavity,namely =Q0(r0,ω)(ω0−ω−i 20)(ωc−ω−i2), the simplest CQED system. Our derivation has the ad- with vantage of relating all CQED parameters to microscopic expressions for the semiconductor QD - nanocavity sys- ω˜ = ω 4πµ2cvωc2a(r0,r0) (14) tem under investigation. 0 0− ~ǫMc2 γ = γ0+γr (15) 8πµ2 ω2b(r ,r ) γr = cv~ǫc c20 0 (16) A. QD susceptibility tensor M g2 = 2πµ2cvωc|Φ0(r0)|2 (17) ~ǫ Insemiconductorswithcubicsymmetry(e.g. InGaAs), M the QD susceptibility tensor is expressed as We then compute the emission spectrum of the system from thelinearresponseequation(11),usingthevirtual oscillating χˆQD(r,r′,ω)= µ~2cvΨ(r)Ψ∗(r′)χQD(ω) 10 01 00  , dtthoipavotalescpumounmettahfinoeedlodu[3fls6ue]c.mtuTisahstieioonmnsies.ttWhhoeedltiihnseerbaearfsoreredespsooonnlvsteehEeofqa.tshs(eu1m1sy)pswttieiotmnh 0 0 0   0(r) given bythefield produced byan oscillating dipole at (9) Q theQD position, in thephotonic structure. Wedefine: whereµ istheBlochpartoftheinterbanddipolematrix cv eQhloeDlme,setnaotknealynn,dahtΨern(rc=)eirstheteh=ezre-lhceo.cmtHrpoeonren-,heonwlteeiwasraeuvneacsfosuuunmpcltieinodgnthioneattvhhyee Sq(ω)= (ωq0−4gω2−−i(γγ2−)4(κω)2c(−ωcω−−ωi−κ2)i−κ2)g2 (18) electromagnetic field. We further assume to deal with Using equation (13), we find: a single QD transition, having one specific polarization (e.g. alongx). Then,thesusceptibilitytensorisreplaced (r,ω)= 0(r,ω)+ Q Q by a scalar, where 1 4c2π~ω√2µǫM2cv 4gS2q(ω(γ)−κ)2 (cid:20)2ωΦc0((ωrc)Φ−∗0ω(r−0)ci2κ2) +gc(r0,r0)(cid:21)Q0(r0,ω) . χ (ω)= . (10) − 4 QD ω0 ω iγ20 q (19) − − The inputfield (r,ω) in our formalism is thefield present 0 Here, γ0 is an additional non-radiative damping rate of inthephotonicsQtructureintheabsenceoftheQD.Thisfield the bare QD resonance. Given the small size of the QD can be computed by a Green’s function procedure similar to withrespecttothecavitymodespatialextension,wecan the one presented above. In this case, following the method safelyapproximateΨ(r)=δ(r r0). Then,Eqs. (6)and ofRef[28],thebackgrounddielectricsystemisthefreespace, − 4 while the perturbation is the photonic structure itself. This IV. BEYOND THE MACROATOM PICTURE procedureispresentedinAppendixA.Wefurtherneglectthe firstandthirdterm ontheright-handsideofEq. 19,asthey Recent studies have demonstrated that a semiconductor are off-resonant with respect to the cavity mode. We obtain QD displays spectral features that depart from the simple thefollowing expression for theemitted field picture of a two-level system. One typical example is the non-perturbative coupling to acoustic phonons, resulting in ǫ(r) (r,ω)= E broadphononsidebandsintheexcitonspectrum. Thismech- p4πωc22~µ2cv 4gS2q−(ω(γ)−4κ)2 (cid:20)2ωΦc0((ωrc)Φ−∗0ω(r−0)ci2κ2)(cid:21)E0(r0,ω) . aiacnnaiilsslmmy [h3th,aas4t,n1ho5wa]sbabneeedenneexrxpetecereninmstivleyneltyianlvclyhesa[tr1iag,ca2tt,eerd5iz].eisdAtbnhooetthhtretarhnmesoietriceohtn-- q (20) between multi-exciton manifolds, involving the continuum of excited states of each manifold (sometimes referred to as This result can be now traced back to the well known ex- “shakeup process”). This mechanism has proven very effec- pressions for the QD-cavity emission spectrum [37, 38]. We tive especially when a QD is embedded in a resonant cavity, usethefactthat (r,ω)isingeneralsmoothlyvaryingasa Q0 givingrisetointensePLatthecavitymodeevenatverylarge functionofω,asdiscussedinAppendixA.Hence,weassume cavity-QDdetuning–thecavity feedingmechanism. Thefor- (r ,ω) . Then E0 0 ≈E0 malism discussed here can be generalized to situations like (r,ω) 2 thefirst one, characterized bya homogeneous spectral modi- E = (r,ω) (ω) , (21) fication, by replacing the simple QD susceptibility (10) with (cid:12) E0 (cid:12) F S the appropriate model. Here, as an example, we discuss the (cid:12) (cid:12) with thesemicon(cid:12)ductor c(cid:12)avity form factor expressed as case of exciton-acoustic phonon coupling with formation of (cid:12) (cid:12) 2 phonon sidebands. We are still interested in determining the (r,ω)= 4πω2µ2cv Φ0(r)Φ∗0(r0)c2 . emission spectrum in theform (21). F (cid:12)(cid:12)c2~ 4g2 (γ−κ)2 (cid:20)2ωc(ωc−ω−iκ2)ǫ(r)(cid:21)(cid:12)(cid:12) (cid:12) − 4 (cid:12) (cid:12)(cid:12) q (cid:12)(cid:12)(22) A. QD susceptibility tensor Theremaini(cid:12)ng factorin Eq. (21)istheemission spectru(cid:12)mof the QD, as in atomic CQED, (ω)= q(ω) 2, expressed in theresonant case (ωc =ω0) asS |S | Thecoupling of oneexciton to theLA-phononband is de- scribedexactly,throughthesolutionoftheindependentBoson (ω)= Ω+−ω0+iκ2 Ω−−ω0+iκ2 2 , (23) model[15,40]. Ithashoweverbeenshown[3]thataverygood S (cid:12)(cid:12) ω−Ω+ − ω−Ω− (cid:12)(cid:12) athcceo2unndt Bofortnhepeexrtcuitrobnatsipoenctlervueml,cwainthbtehoebatdavinaendtaaglereoafdhyaavt- with (cid:12) (cid:12) (cid:12) (cid:12) ing a simple expression for the exciton-phonon self-energy. i γ κ 2 Ω± =ω0− 4(γ+κ)±rg2− −4 . (24) Wexecittohnu-sprheownrointesetlhfeenQeDrgyexacsiton susceptibility including the Thisis theusualCQED result [37,39(cid:16)]. Wes(cid:17)eefrom equa- 1 tion (14) that thecoupling of the QD to theelectromagnetic χQD(ω)= ω ω iγ0 +Σ(ω), (26) field of the modes other than the cavity mode, produces a 0− − 2 radiative shift and an additional radiative damping, respec- where, within second Born approximation and restricting to tively proportional to the real and imaginary parts a(r0,r0) only one phonon band, andb(r ,r )ofthephotonGreen’sfunction. Theshiftsimply 0 0 redefines theresonant frequency and will be neglected in the Σ(ω)= |gqx|2(1+n(q)) + |gqx |2(n(q)) . following. Thebackgroundelectromagneticfieldhowever,has q "ω+iγ2 −ω0−ω(q) ω+iγ2 −ω0+ω(q)# alsoanimpactontheradiativedampingoftheQD.Inpartic- X umlaord,eγlsrt=he4rπaµd2cv~iaωǫMtc2ibvc(2er0d,re0c)a,yoriingtionatthinegnofrnormesothneantterbmacbk(grr0o,ur0n)d, Htioenre,atn(tqe)mipsetrhateuBreoskeB-ETi.nsWteeincoenqsuiidliebrrituhme cpahseonoofndoefcocrumpaa-- tion potential coupling with acoustic phonons of dispersion electromagnetic field. Within the CQED formalism, the QD decayratedenotedbyγcanbelinkedtoourresultbydefining ωq = qs (q =| q |), where s is the sound velocity, as in Ref. 15. In Fig. 1 we display the imaginary part of the QD sus- theFodramexpaimngplrea,teweγu=seγ0E+q.γr(.14) to compute the Rabi split- ceptibility, as computed at kBT = 10K for an InAs QD of 10nm diameter. It should be noted that the phonon spectral ting of a semiconductor QD embedded in a photonic crystal featuresdonotdependspecificallyontheshapeoftheexciton nanocavity. We model the optical cavity mode as a Gauss- wavefunctionbutare onlydetermined byitsvolume[15]. In shaped mode Φ (r) with spatial extension corresponding to 0 theplot,wenoticethepronouncedsidebandscomparedtothe thetypical size of a mode in this system. Wefurther assume spectrum of an ideal exciton. The sidebands are more pro- zero QD-cavity detuning, and the QD position centered at nounced on the high energy side, where they are determined r0 =0. By defining the volume Vm of the Gauss mode, we byacoustic phonon emission. find: g2= 2πµ2cvωc . (25) ~ǫ V M m B. Emission spectrum ThisexpressioncoincideswiththatobtainedinRef. 37. With realistic numerical values for InAs QDs in photonic crystal nanocavities (µ2cv = 480meVnm3, Vm = 0.04µm3), we find Intuitively, the emission intensity at the cavity-mode fre- g 200µeV. quency depends on the optical density of the underlying ex- ≈ 5 2 10 T=10K cavity mode 0 Phonons 2 (a) 101 E|0 5 /| 2 )] ω)| χω(100 E(r, m[ | 0 I 10−1 2 (b) |0 E 5 /| 2 )| 10−2 ω 1060 1062 r, ω(meV) E( | 0 FIG. 1: Imaginary part of the quantum dot susceptibility in presence of LA-phonon coupling (full) and without phonons 2 (c) (dashed), computed at T=10K. As an illustration, we plot E|0 5 the cavity mode optical density at 1 meV positive detuning 2/| (dotted). ω)| r, ( E | 0 citonspectrum. Hence,thepresenceofacousticphononside- bandsisexpectedtoenhancethisPLintensity,whenthecav- ity is detuned from the exciton. This is illustrated in Fig. 1, 2 (d) |0 wherethecavitymodespectrumisplottedat1meVpositive E 5 detuningwith respect to theexciton peak. 2/| WeusetheQDsusceptibility(26)tocomputetheemission ω)| spectrum (21). The form factor (r,ω) is still expressed as r, (22),while theQD emission spectFrum now reads |E( 0 0 2 (ω)= 4g2− (γ−4κ)2(ωc−ω−iκ2) 2. (27) ω−ω0(meV) S (cid:12)(ω0−qω−iγ2 +Σ(ω))(ωc−ω−iκ2)−g2(cid:12) (cid:12) (cid:12) As expecte(cid:12)d, the exciton-phonon coupling results in(cid:12) a mod- ified emission spectrum. In particular, the exciton-phonon FIG. 2: Plot of the emission spectrum of the QD-cavity sys- self energy is responsible for a modified intensity at the cav- tem in the presence of phonons (solid line) for different de- ity mode frequency and a small polaron shift of the exciton tunings D = ω0 ωC. a) D = 1meV, b) D = 0meV, c) − − frequency. In Fig. 2, we plot the computed spectrum at D=1meV,d) D=2meV (kBT=10K).Comparison with no kBT = 10 K for various values of the exciton-cavity detun- phonons(dashed line) is also given. ing. While the strong coupling features remain essentially unchangedforzero-detuning(seepanelb),wecanclearly see in panels (a), (c), and (d) that phonon sidebands can effi- ciently emit through thecavity mode, as also found byother theoretical approaches [26, 41]. As a consequence, the peak at frequency ω ωc is enhanced with respect to the simple ≈ CQED model, provided the detuning is not larger than the energy extent of the phonon bands. This effect was widely ity mode, observed in several recent experiments, could be investigatedexperimentallyduringthelastyears. Ithasbeen attributedtothephononsidebandmechanism. Wepointout reported for QDs embedded in various systems such as pho- however, that this cavity feeding phenomenon has been ob- tonic crystal nanocavities [13, 21, 42] and micropillars [24]. served also when the exciton-cavity detuning is much larger We can see in Fig. 2(d) that for detuningexceeding the typ- than the typical width of the phonon broadbands, ie a few ical broadband width, this feature starts to disappear, as at meV. These observations are accompanied by a superlinear large detuning the sideband essentially vanishes. Moreover, dependence of the cavity mode PL on the excitation power. thestrongasymmetryinχˆQD(ω)atlowtemperatureshasfor The phonon sideband mechanism, on the other hand, is ex- consequencethatthepersistenceofthepeakissmallforneg- pected to provide a spectrum that depends linearly on the ativedetuning(seepanel(a)). Finally,inFig. 3,weshowthe excitation power. The phonon sideband model is thus ex- influenceof the temperatureon theemission spectrum S(ω). pectedtoholdmostlyatsmalldetuning. Theobservationsof The phonon sidebands grow with temperature and result in cavityfeedingatlargerdetuningaremostlikelyduetomulti- an increased emission through the cavity mode. exciton emission, partially involving wetting layer states, as Thisresultsuggeststhattheenhancedemissionatthecav- has been recently discussed [20, 21, 43, 44]. 6 the strong coupling by compensating for the cavity losses – a mechanism that can be traced back to the gain produced 0 Phonons T=4K by the excitation of the additional QDs in the cavity. This T=10K T=40K enhancementcannotbereproducedbyourmodel,asitwould require accounting for nonlinear optical response. Eq. (28) 2| 4 thus can model the presence of spectator QDs only in the 0 E limit of very small average population of each QD,for which /| 2 thelinear assumption holds. )| ω r, E( 2 | 0 0 1 ω−ω0(meV) FIG. 3: Emission spectrum of the QD-cavity system plotted fordifferentvaluesofkBT (arbitraryunits),ωc ω0=1meV. − C. Influence of neighboring QDs One major assumption of this model is that there is only one QD dot located within the region in which the cavity mode is extending. Given the density of the QD ensemble and the spatial extension of the nanocavity mode, it might well be that spectator QDs – i.e. additional QDs present in thecavity–contributetotheemission spectrum. TheseQDs aremost likelyweaklycoupledtothecavitymodebecauseof strong energy detuningor of smaller spatial overlap with the mode wave function. It has been recently suggested [45, 46] that,iftheseQDsareexcitedinadditiontothemainQD,the resulting emission spectrum is substantially modified, some- times even leading to a recovery of strong coupling in a sit- uation that would be of weak coupling if only the main QD was excited. In Refs. 45 and 46, this effect has been mod- eled by an additional pump term acting on the cavity mode. Here,we can account forthepresenceof additional QDsin a naturalway,bygeneralizing theexpression forthesingle-QD susceptibility (9). Thenew susceptibility then reads χˆQD r,r′,ω = µ~2cv 10 01 00 V. CONCLUSION 0 0 0 (cid:0) (cid:1)   × "Ψ(r)Ψ∗ r′ χQD(ω)+ Ψj(r)Ψ∗j r′ χj(ω)#(2,8) j (cid:0) (cid:1) X (cid:0) (cid:1) where 1 χj(ω)= γ . (29) ωj ω i 0j +Σj(ω) − − 2 Inconclusion,wehaveshownthattheGreensfunctionfor- Here, the j-th QD has parameters defined analogously to malism is a powerful tool to relate quantitatively the usual those of the main QD. Starting from this expression, the atom-CQED parameters to the description of any QD-cavity derivation of the emission spectrum can be carried out anal- system. We also extended this formalism to a QD weakly ogously to the single-QD case. In particular, when deter- coupled to LA phonons. Thus we underlined that the differ- mining the input field (r,ω) as described in Appendix A, enceofaQDtoasimpletwolevelsystemisgreatlyenhanced 0 Q the virtual oscillating dipole will consist of a sum of terms whenthequantumdotisplacedinsideananocavity. Itmakes originating from the different QDs, with relative weights Bj possiblethataPLpeakofconsiderableamplituderemainsat that express the contribution of each QD to the initial state cavityfrequencyevenforlargedetuningcomparedtotheRabi of the emission process. It should be pointed out however, splitting,butlimitedtoafewmev. 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Fiore (2009), http://link.aps.org/abstract/PRB/v79/e235325. arXiv:cond-mat/0907.3392. 8 Appendix A: Determination of the bare cavity which is indeed a plane wave. At this point, we will use the electric field using the virtual oscillating dipole method [31]. We replace B by a point source centered in r0: B(r,ω)= Bδ(r r0). E E − Using thebackground Green’s function defined as: To determine (r,ω), we will follow the approach pro- 0 Q posed in Ref. 28. We have ω2 ′ ∇∧∇∧GB(r,ω)− c2ǫBGB(r,ω)=δ(r−r), = ǫ(r) (r,ω) (A1) 0 0 Q E with p we have: 0(r,ω)= B(r,ω) (A2) ∇∧∇∧E0(r,ω)− ωc22ǫ(r)E0(r,ω)=0 +E dr′GBE(r,r′,ω)ωc22∆ǫ(r′)EB r′,ω (A3) ZV That is, with ∆ǫ(r)=ǫ(r) ǫB, ω2(cid:0) (cid:1) − =B δ(r−r0)+GB(r,r0,ω)c2∆ǫ(r0) . (A4) (cid:20) (cid:21) ∇∧∇∧E0(r,ω)− ωc22ǫBE0(r,ω)= ωc22∆ǫ(r)E0(r,ω) In this expression, GB(r,r′,ω) is a slowly varying and espe- cially non-resonant function of ω. Then so does (r,ω) and 0 E Wedefine B(r,ω) as a solution of finally: E (r,ω)= ǫ(r) (r,ω), (A5) 0 0 ω2 Q E ∇∧∇∧EB(r,ω)− c2ǫBEB(r,ω)=0, with Q0(r,ω) a function withpno resonance.