Published: May 02, 2011

r 2011 American Chemical Society 2318 dx.doi.org/10.1021/nl200579f |Nano Lett. 2011, 11, 2318–2323

LETTER

pubs.acs.org/NanoLett

Quantum Plexcitonics: Strongly Interacting Plasmons and ExcitonsA. Manjavacas,*,† F. J. García de Abajo,† and P. Nordlander‡

†Instituto de �Optica - CSIC, Serrano 121, 28006 Madrid, Spain‡Department of Physics and Astronomy, M.S. 61, Rice University, Houston, Texas 77005-1892, United States

bS Supporting Information

The study and design of devices capable of controlling light-matter interaction at the nanoscale have been subjects of

intense activity over the past decade.1Metallic nanostructures areideally suited to this end due to their ability to focus and trapoptical energy into subwavelength spatial regions.2 The largefields and the high confinement associated to the plasmonicresonances supported by these systems3 enable strong interac-tions with other photonic elements such as quantum emitters.4,5

Thanks to the advance in nanofabrication techniques, we arecurrently approaching length scales in which the quantumbehavior of these structures becomes important, thus openingwide horizons for new designs and applications in the novel areaof quantum plasmonics.6

Understanding the physics underlying the internal interactionin these systems is important to take advantage of their quantumfeatures, such as collective and single-particle excitations,7 andquantum correlations and interferences.8 The proper theoreticalcharacterization of these processes thus requires a fully quantummechanical framework. Numerical simulations of Maxwell’sequations, which have been successfully used to model theoptical response of a vast number of systems involving metallicnanostructures,9 cannot describe completely the interferenceand couplings between quasi-particles such as plasmons andexcitons. It is therefore important to develop new theoreticaltools that overcome such limitations. In this direction, somepreliminary results have been recently reported based upon thedensity matrix formalism.10,11

In this paper, we show that the Zubarev’s Green functions12

method provide a convenient approach for modeling the opticalresponse of plasmons interacting with quantum emitters. Thispowerful approach enables us to describe the internal evolutionof such quantum systems beyond the perturbative regime.

The major advantage of this approach is that it can be system-atically extended to include a range of increasingly complexphenomena. For instance, by directly including the couplingbetween the plasmons and a continuum of electronic states, Fanoresonances emerge in a natural way.13,14

The application of this formalism is illustratedwith two differentexamples involving hybrid plexcitons (i.e., systems formed by theinteractions of plasmonic nanoparticles with nearby excitonicsystems such as quantum dots or molecules). In a first instance,we consider a simple situation in which a quantum emitter isplaced close to the surface of a metallic nanoparticle. In a secondexample, we describe a quantum emitter placed at the gap of ananoparticle dimer.15 This type of structure is currently attractingmuch attention16,17 due to its potential application as a platformfor quantum information devices.Zubarev’s Green Functions and Their Application to

Nano-Optics. The method of Zubarev’s Green functions12 hasbeen successfully applied to different problems in statisticalphysics and linear response theory.18�20 Here, we adapt it tostudy the optical absorption properties of hybrid systems formedby plasmonic structures and quantum emitters (e.g., molecules orquantum dots). In particular, we compute absorption spectraobtained from the retarded Zubarev Green function of thequantum operators that mediate the photon absorption process.The time-domain Green function of two of these operatorscontains the information on the evolution of the quantumprocess that they represent, and its Fourier transform is thereforedirectly related to the absorption spectrum.

Received: February 18, 2011Revised: April 18, 2011

ABSTRACT:We present a fully quantummechanical approachto describe the coupling between plasmons and excitonicsystems such as molecules or quantum dots. The formalismrelies on Zubarev’s Green functions, which allow us to gobeyond the perturbative regime within the internal evolutionof a plasmonic nanostructure and to fully account for quantumaspects of the optical response and Fano resonances in plas-mon�excition (plexcitonic) systems. We illustrate this methodwith two examples consisting of an exciton-supporting quantum emitter placed either in the vicinity of a single metal nanoparticle orin the gap of a nanoparticle dimer. The optical absorption of the combined emitter�dimer structure is shown to undergo dramaticchanges when the emitter excitation level is tuned across the gap-plasmon resonance. Our work opens a new avenue to deal withstrongly interacting plasmon�excition hybrid systems.

KEYWORDS: Quantum optics, plasmon, exciton, plexciton, fano resonance

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The retarded Zubarev Green function of two operators A andB is defined in the frequency domain as

ÆÆA; Bææω þ i0þ ¼ � ip

Z ¥

�¥dteiðω þ i0þÞtθðtÞƽAðtÞ, Bð0Þ�ηæ

ð1ÞwhereA(t) is theHeisenberg representation21 of operatorA,θ(t)is the Heaviside step function, and the brackets [A,B]η = AB �ηBA stand for the commutator of bosonic operators (η = 1) orthe anticommutator of fermionic operators (η = �1). Noticethat an infinitesimally small positive imaginary part is added tothe frequency in order to ensure the convergence of the integral.The optical absorption spectrum is then related to the retarded

Zubarev Green function as

σðωÞ � � ImfÆÆA;Aþææω þ i0þg ð2Þ(a derivation of this expression is given in the SupportingInformation), where A (Aþ) is the annihilation (creation) opera-tor of an excitation in the system, resulting from the emission(absorption) of one photon. This type of Green function iscommonly calculated by writing its equation of motion

pωÆÆA;Aþææ ¼ ƽA,Aþ�ηæþ ÆƽA,H �;Aþææ ð3Þ(see the Supporting Information for a derivation of this equation),which depends on another Green function ÆÆ[A,H];Aþææ , whereH is the Hamiltonian. In a similar way, this new Green functioncan be also calculated by writing down its equation of motion.Iterating this process, one obtains a hierarchy of equations thatmay need to be truncated at some point by applying a physicalapproximation. Finally, a linear system of equations is derived,from which ÆÆA;Aþææ, and therefore the optical absorption spec-trum, is obtained.A Quantum Emitter Interacting with a Metal Nanoparti-

cle. We consider a simple but illustrative system composed of asmall metallic nanoparticle with a diameter of tens of nanometersand a quantum emitter (e.g., a quantum dot22,23 or a conjugatedpolymer molecule24) placed in the vicinity of the nanoparticle.Such a coupled plasmon-exciton system will result in hybridplasmonic excitonic modes also referred to as Plexcitons.25 Weassume that the nanoparticle can support a well-defined bosonicdipolar plasmonmode and neglect higher-multipole modes. Thisis a reasonable approximation for a noble-metal particle with aradius much smaller than the wavelength. Additionally, wedescribe the quantum emitter as a fermionic system with onlytwo possible states (ground and excited).The Hamiltonian describing the noninteracting evolution of

the fermion and the plasmon is

H 0 ¼ εddþdþ εcc

þc ð4Þwhere d and c (dþand cþ) are the annihilation (creation)operators for the nanoparticle plasmon and the quantum emitterfermion of energies εd and εc, respectively. The plasmon-fermioninteraction is modeled by the Hamiltonian

H int ¼ �Δdc½dþcþ cþd� ð5Þwhere Δdc is the plasmon-quantum emitter coupling constant,which we take to be real.26

In a realistic model, one has to describe the finite lifetime of theexcitations of the system, which produce finite widths in thecorresponding spectral resonances. Generally, such widths are

the result of the inelastic interactionwith a continuumofmodes. Forinstance, a plasmon can decay radiatively by emitting one photonand nonradiatively through the generation of electron�holepairs, phonons, etc. We describe these inelastic interactions byadding a term to the Hamiltonian

H decay ¼Z

dωpωfþd ðωÞfdðωÞ þZ

dωpωfþc ðωÞfcðωÞ

�Z

dω½vdðωÞfdðωÞdþ þ v�dðωÞfþd ðωÞd� �

Zdω½vcðωÞfcðωÞcþ

þ v�c ðωÞfþc ðωÞc� ð6Þ

where fd(ω) and fc(ω) are the annihilation operators of thecontinuum modes that couple to the plasmon and fermion,respectively. The corresponding coupling constants are vd(ω)and vc(ω). These terms account for all possible decay channels ofthe plasmon and the quantum emitter. The total Hamiltonian ofour system is given by the sum H = H0 þ Hint þ Hdecay.Assuming that only the plasmons couple efficiently to the

external photons, the optical absorption spectrum can be foundfrom the Green function ÆÆd;dþææ [see eq 2]. This should be agood approximation in view of the fact that the absorption crosssection of the nanoparticle is generally much larger than that of amolecule or a quantum dot. Using eq 3 and the Hamiltonianderived above, the equation of motion for ÆÆd;dþææ becomes

ðpω� εdÞÆÆd; dþææ ¼ 1�ΔdcÆÆc; dþææ�Z

dω0vdðω0ÞÆÆfdðω0Þ; dþææ

ð7ÞFrom this expression, it is clear that we need to compute twoadditional Green functions: ÆÆc;dþææ and ÆÆfd(ω);dþææ. The firstof them can be easily obtained from its equation of motion

ðpω�εcÞÆÆc; dþææ ¼ �ΔdcÆÆð1� 2cþcÞd; dþææ�Z

dω0vcðω0ÞÆÆð1� 2cþcÞfcðω0Þ; dþææ ð8Þ

Therefore, we see that new Green functions emerge that need tobe calculated. The iteration of this process would produce aninfinite hierarchy of equations ofmotion, but we truncate it at thispoint by approximating the operator cþc by its expectation valueÆcþcæ = nc. The presence of this term is the result of the fermioniccharacter of the quantum emitter.We still need to deal with Green functions containing infor-

mation related to the inelastic decay processes (i.e., ÆÆfd(ω);dþææ,and ÆÆfc(ω);dþææ). Upon inspection of the Hamiltonian, weconclude that these Green functions only depend on thosederived above, and they are given by

ðpω� pω0ÞÆÆ fdðω0Þ; dþææ ¼ � v�dðω0ÞÆÆd; dþææ

ðpω� pω0ÞÆÆ fcðω0Þ; dþææ ¼ � v�c ðω0ÞÆÆc; dþææ ð9Þ

Substituting the first of these expressions into eq 7 and introdu-cing the positive infinitesimal imaginary part of the frequencythat appears in the definition of Zubarev’s Green function [seeeq 1], we find the integralZ

dω0 jvdðω0Þj2pω0 � pω� i0þ

¼ PZ

dω0 jvdðω0Þj2pω0 � pω

þ iπjvdðωÞj2

¼ δωd þ iΓd

2ð10Þ

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which we have evaluated using the Sokhatsky�Weierstrass theorem.Here, P denotes the Cauchy principal value, δωd represents afrequency shift, andΓd = 2π|vd(ω)|

2 is the inelastic decay rate of thedipolar plasmons. Dealing in a similar way with the remainingfermionic decay channels, we can transform eqs 7 and 8 into

pω� εd þ δωd þ iΓd

2

� �ÆÆd; dþææ ¼ 1�ΔdcÆÆc; dþææ ð11Þ

and

pω� εc þ ð1� 2ncÞ δωc þ iΓc

2

� �� �ÆÆc; dþææ ¼ �Δdcð1� 2ncÞÆÆd; dþææ

ð12Þwhich can be easily solved to obtain the desired Green function ÆÆd;dþææ. Finally, inserting this result into eq 2, we obtain the opticalabsorption spectrum of the plasmon-fermion system

σ� Im pω� εdþ δωd þ iΓd

2� Δ2

dcð1� 2ncÞpω� εc þ ð1� 2ncÞ δωc þ i

Γc

2

� �8>>><>>>:

9>>>=>>>;

�1

ð13ÞThis equation clearly shows that the plasmonic resonance ismodified by the interaction with the quantum emitter. However,although the fields near a metallic nanoparticle can be large dueplasmonic enhancements, any realistic value of the couplingconstant Δdc of this system is too small to induce an observablechange of the spectrum. A noticeable effect can be nonethelessobserved when the emitter is situated in a hot spot of a morecomplex plasmonic structure, such as the gap of a particle dimerin which a well-defined single gap mode can be clearly resolvedfor small gap distance.We consider this situation later in this paper,but first we describe the more complex interaction between theemitter and the plasmons in the two nanoparticles of the dimer.A Quantum Emitter in a Metallic Gap. We now move to a

more complicated system in which the interaction between thequantum emitter and the plasmon is largely enhanced, thusproducing noticeable effects in the optical absorption spectrum.More precisely, we consider two identical metallic nanoparticleswith diameter of tens of nanometers separated by a gap of a fewnanometers. A quantum emitter is placed in this gap, where thefield-amplitude enhancement due to the plasmonic resonances ofthe particles can easily reach 2 orders of magnitude with respect tothe value of an external plane wave.3 The optical response of thissystem has been recently studied using classical electrodynamicsmethods.16,17 We assume each particle to be supporting just onedipole plasmon and one quadrupole plasmonwithm= 0 azimuthalsymmetry with respect to the dimer axis. Only modes with thissymmetry can be excited because we consider the external field tobe oriented along the dimer axis. In a real system, higher-ordermultipoles are also contributing for small separations, although thedipoles are dominant for the separations discussed below. In fact,the quadrupole can be considered to effectively account for theeffects of all higher-order modes combined. The quantum emitteris described as a two level system like in the previous section.The Hamiltonian that determines the behavior of this nanostruc-ture contains again three terms H = H0 þ Hint þ Hdecay. Thenoninteracting part reads

H 0 ¼ εd½dþ1 d1 þ dþ2 d2� þ εq½qþ1 q1 þ qþ2 q2� þ εccþc ð14Þ

where di and qi are the dipolar and quadrupolar plasmon bosonicannihilation operators of particle i= 1,2, respectively. Furthermore,εq is the quadrupolar plasmon energy, while the rest of theoperators have a similar meaning as in eq 4. The interactionHamiltonian Hint must describe all possible couplings of thedifferent system excitations. In our particular case, the dipolarand quadrupolar plasmons of one particle can interact with theexcitations of the other one via dipole�dipole (Δdd), dipo-le�quadrupole (Δdq), and quadrupole�quadrupole (Δqq) cou-pling constants. In addition, the quantum emitter interacts withboth the dipole plasmons (Δdc) and quadrupole plasmons (Δqc).Therefore, we have

H int ¼ �Δdd½dþ1 d2 þ dþ2 d1� �Δqq½qþ1 q2 þ qþ2 q1� �Δdq½dþ1 q2 þ qþ2 d1��Δdq½dþ2 q1 þ qþ1 d2� �Δdc½dþ1 cþ cþd1� �Δdc½dþ2 cþ cþd2��Δqc½qþ1 cþ cþq1� �Δqc½qþ2 cþ cþq2� ð15Þ

Finally, the decay part of the Hamiltonian is given by eq 6, wherewe now need to sum over both dipole plasmons and include extraterms for the quadrupole plasmons.For particles with diameters of tens of nanometers, the

quadrupoles couple only weakly to the light. As in the previoussection, we can neglect the direct coupling of the emitter toincident light. We therefore assume that only the dipolarplasmons couple efficiently to the external photons. (However,the extension of our calculations to include the direct coupling tothe emitter and the quadrupoles is straightforward.) With thisapproximation, we just need to obtain the Green function ÆÆD;Dþææ, where D is a linear combination of the dipolar plasmonannihilation operators of particles 1 and 2 compatible with thesymmetry of the external field. We choose D = d1 þ d2, whichimplies that the incident photons are absorbed at the twoparticles with the same phase. This represents the situation inwhich the external field is incident perpendicularly to the dimeraxis, and therefore, only the bonding mode15 of the dimer isexcited.Using eq 3 and theHamiltonian derived above, the equation of

motion of the Green function ÆÆD;Dþææ reads

ðpω�εd þΔddÞÆÆD;Dþææ ¼2�ΔdqÆÆQ ;Dþææ� 2ΔdcÆÆc;Dþææ

�Z

dω0vdðω0ÞÆÆ fDðω0Þ;Dþææ ð16Þ

whereQ = q1þ q2 and fD = fd1þ fd2. Like in the previous section,we are encountering additional Green functions. In particularÆÆQ;Dþææ, whose equation of motion is

ðpω�εq þΔqqÞÆÆQ ;Dþææ ¼ �ΔdqÆÆD;Dþææ� 2ΔqcÆÆc;Dþææ

�Z

dω0vqðω0ÞÆÆ fQ ðω0Þ;Dþææ ð17Þ

with fQ = fq1þ fq2. Similarly, the dynamics of ÆÆc;Dþææ is governedby

ðpω�εcÞÆÆc;Dþææ ¼ �Δdcð1� 2ncÞÆÆD;Dþææ�Δqcð1� 2ncÞÆÆQ ;Dþææ

� ð1� 2ncÞZ

dω0vcðω0ÞÆÆ fcðω0Þ;Dþææ ð18Þ

where we have truncated the hierarchy of equations of motion byreplacing the operator cþc in the Green functions by its expecta-tion value nc. We also need to follow three additional Greenfunctions involving the operators fD, fQ, and fc, which appear inthe equations of motion derived above. Examining the Hamilto-nian, it is clear that they are given by expressions similar to eq 9,

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which can be immediately inserted into eqs 16�18. Then,following the same procedure as in the previous section to dealwith the integrals over frequency [see eq 10], we end up with alinear system of three equations and three unknowns: ÆÆD;Dþææ,ÆÆQ;Dþææ, and ÆÆc;Dþææ. Finally, solving this system of equationswe obtain ÆÆD,Dþææ, and therefore, via eq 2, the optical absorp-tion spectrum

σ � Im Sd � 2Δ2dcð1� 2ncÞ

Sc�

Δdq �2ΔqcΔdcð1� 2ncÞ

Sc

" #2

Sq �2Δ2

qcð1� 2ncÞSc

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

�1

ð19Þwhere we have defined Sd = pω�εd þ Δdd þ δωd þ iΓd/2, Sq =pω�εqþΔqqþ δωqþ iΓq/2, and Sc = pω�εcþ (1�2nc)[δωc

þ iΓc/2].In order to apply this model to actual dimer-quantum-emitter

systems, we need to find realistic values for the different para-meters involved in eq 19. We illustrate this by considering twosilver nanoparticles of 10 nm radius separated by a gap of 4 nm.We take the system to be initially prepared in the ground state, sothat nc = 0 for the emitter placed in the dimer gap. Moreover, thefrequency shifts δωd and δωq can be accounted for by renor-malizing the energies of the corresponding states εd, and εq,respectively. Notice that we keep δωc as it cannot be reabsorbedwhen nc 6¼ 0. We extract the energies and widths of the dipolarand quadrupolar plasmons from classical electromagnetic simu-lations of the absorption cross section for a single silver nano-particle usingMie theory. In a similar way, we obtain the couplingconstants describing the interaction between the dipolar andquadrupolar plasmons by fitting eq 19 with Δdc = Δqc = 0 (i.e.,without emitter) to an electromagnetic simulation of the dimerabsorption cross section using a multiple-scattering approach.27

The resulting values for the different parameters are given inFigure 1b.Figure 1 shows the optical absorption spectrum of the system

under study, computed from eq 19, for different energies of thequantum emitter resonance. The width of this resonance hasbeen taken to be Γc = 4 meV, which is a realistic value for actual

quantum dots.28 It is important to note that we have consideredquantum emitters of resonance energy lying close to the dimerplasmons. For a given emitter energy within the 3�3.5 eV range,the dimer plasmon frequencies can be easily tuned by selectingthe size, shape, and separation of the nanoparticles29 (e.g., usingnanoshells of appropriate thickness30).We have calculated the strength of the coupling between the

quantum emitter and the dipolar and quadrupolar plasmons, andthe value of δωc by fitting eq 19 to a simulation performed withclassical electromagnetic theory, as shown in the SupportingInformation. In this model the quantum emitter is represented bya point dipole with a linear polarizability31R(ω) =μ2/(εc� pω�iΓc/2), where μ is the transition dipole. In our particular examplewe assume μ/e = 0.3 nm, which is a realistic value for a quantumdot.32 Using this transition dipole we obtain the values for thedifferent model parameters shown in Figure 1b. Thanks to thelarge plasmonic fields existing in the dimer gap, this systemexhibits large values of the coupling constants Δdc and Δqc ascompared to the case with a single particle.As we observe in Figure 1, the optical spectra exhibit three

distinct features. The hybridized bonding dipolar and quadru-polar dimer modes around 3.3 (≈ εd�Δdd) and 3.5 eV (≈ εq�Δqq); and the excitonic mode, which is clearly visible as a Fanoresonance with a line shape that strongly depends on the energyεc. The Fano resonance results from the interaction between acontinuum of modes and a narrow discrete mode.14 In ourexample, the quantum emitter resonance is the narrow mode,while the dipolar and quadrupolar plasmons, with their largerwidths, play the role of a continuum. As expected these reso-nances can be fitted approximately using the conventional Fanoline shape with |q|, 1. For instance, in the case of the curve ofFigure 2b corresponding to εc = 3.35 eVwe have q∼ 0.07� 0.05.In Figure 2, we show how the width of the exciton influences

the Fano resonance. The resonance width is an essential factor toobserve Fano resonances in this type of system. Figure 2 alsoshows that the Fano resonance displayed for small values of Γc

disappears as this parameter approaches the value of the dipolarplasmon width Γd.In Figure 3, we illustrate the critical role played by the strength

of the coupling between the quantum emitter and the dipolarplasmon Δdc in determining the shape of the optical absorptionspectrum. For small values ofΔdc, the dip associated to the Fano

Figure 1. Optical response of a hybrid plexcitonic system formed by a quantum emitter placed in the gap of a nanoparticle dimer. (a) Description of thesystem under study. (b) Values of the different model parameters (see text). (c) Energy diagrams of the quantum emitter (fermionic character), and thedipolar and quadrupolar plasmons supported by the nanoparticles (bosonic character). (d) Optical absorption spectrum of the dimer�quantum�emit-ter system, computed for different values of the emitter resonance energy. A Fano resonance is clearly observed as a result of the coupling between theemitter and the particle plasmons.

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resonance almost disappears. For increasing Δdc, the dip be-comes deeper and the separation between the two resultingpeaks becomes more pronounced. Actually, when this couplingconstant is larger than the dipolar plasmon half width Γd/2 wecan alternatively interpret the resulting line shape as vacuumRabisplitting.16

Nonlinear Effects and Beyond. It is important to stress thatour results reproduce the simulations performed in previousworks.16,17 However, our methodology allows us to go one stepforward in the understanding of the physical mechanismsgoverning the response of the system. First of all, we can clearlyidentify the origin of the peaks and other features that character-ize the investigated optical absorption spectra. Furthermore, ourmodel can capture the nonclassical behavior of the differentelements of the system. For instance, in a situation in which thequantum emitter is not initially prepared in its ground state, andhence, the expectation value nc does not vanish. In such case, theoptical response of the system changes dramatically [see eq 19].In particular, under strong optical pumping this expectation valueshould be∼1/2, for which the interaction between the quantumemitter and the dipolar plasmon disappear, leading to an inhibi-tion of the Fano resonance. Such nonlinear Fano resonanceshave been observed in quantum-well structures and are of clearfundamental importance.4,33 This can be clearly observed inFigure 4, which shows the dependence of the optical absorption

spectrum on the expectation value nc, for different emitterexcitation energies εc. As nc grows from 0 to 0.5 the Fanoresonance becomes weaker and finally disappears, reemergingagain for higher values. At the same time, the position of the Fanoresonance is shifted toward higher energies. This behavior, whichis associated to the saturation of the quantum emitter, emergesfrom the fermionic nature of this system. For this reason,although it could be modeled a posteriori using a classical or asemiclassical approach by assuming a change in the oscillatorstrength, it could not be predicted a priori, unless a fullyquantum approach is employed. In practical terms, saturationcan be obtained with a pumping-light frequency differing fromthe probing frequency, but separated from it by less than thewidth of the emitter excitation. The nonlinear response asso-ciated to this nonclassical behavior opens the possibility ofexternally controlling the optical absorption spectrum of thesystem.The Zubarev’s Green functions12 method presented in this

paper is conceptually much simpler than standard Green’sfunction or density matrix approaches since the only formalismthat is required is the evaluation of simple commutation relationsbetween operators. However, the major advantage of the presentapproach is that all dynamics are obtained nonperturbativelydirectly from the Hamiltonian. Because of this, by simply addingmore terms to the Hamiltonian, it is straightforward to includemore complex interactions such as the coupling to the con-tinuum which introduced the broadening of the plasmon andexciton states already discussed. Most importantly, our methodprovides a simple approach for including much more compli-cated effects such as strong correlation, coupling between twoemitters, coupling to phonons and other collective excitations.For instance, an interesting scenario is presented whentwo quantum emitters are placed at the gap of a particle dimer.The interaction between emitters can be affected by Coulomb

Figure 4. Dependence of the optical absorption spectrum on theexpectation value of the emitter excited level nc for different values ofthe emitter excitation energy. The rest of themodel parameters are givenin Figure 1b. The absorption spectrum changes dramatically as ncchanges from 0 to 1.

Figure 2. Dependence of the optical absorption spectrum on thequantum emitter resonance width Γc. We take the emitter excitationenergy εc = 3.35 eV, while the rest of the model parameters are given inFigure 1b. The Fano resonance disappears as Γc increases.

Figure 3. Dependence of the optical absorption spectrum on thestrength of the coupling between the dipole plasmons and the quantumemitterΔdc.We take the emitter excitation energy εc = 3.35 eV, while therest of the model parameters are given in Figure 1b. A dramaticdependence of the spectral shape on Δdc is observed.

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repulsion, which can be incorporated into our formalism byadding a Hubbard term to the Hamiltonian as it is commonlydone in the Anderson model.34 Such a term prevents the twoquantum emitters from simultaneously lying in excited states andmay therefore introduce additional useful tunable nonlinearbehavior of relevance for plexciton-based quantum informationdevices.Conclusions. In summary, we have presented a new approach

based on the Zubarev’s Green functions method for the study ofthe optical properties of plasmonic systems interacting withquantum emitters (e.g., molecules or quantum dots). Theformalism describes the quantum internal evolution of suchfermionic-bosonic hybrid systems beyond the perturbative re-gime. Its power is here demonstrated with two illustrativeexamples: (1) a system consisting of a quantum emitter placedclose to a metallic nanoparticle and (2) a quantum emitter placedin the gap of a nanoparticle dimer. Using realistic parameters, weshow that the optical absorption spectrum of the dimer-quan-tum-emitter system can exhibit Fano resonances resulting fromthe interaction between the quantum emitter and the gapplasmon, in agreement with previous work.16,17 Our approachcan be straightforwardly generalized to include more compli-cated interactions. The formalism is likely to become a powerfultool for the description of plasmon�excition interactions inplasmonic transistors, modulators, and quantum informationdevices.

’ASSOCIATED CONTENT

bS Supporting Information. An extra figure related to thedetermination of the model parameters and the derivations ofeq 2 and eq 3 are provided. This material is available free ofcharge via the Internet at http://pubs.acs.org.

’AUTHOR INFORMATION

Corresponding Author*E-mail: a.manjavacas@csic.es.

’ACKNOWLEDGMENT

This work has been supported by the Spanish MICINN(MAT2010-14885 and Consolider NanoLight.es) and the Eur-opean Commission (FP7-ICT-2009-4-248909-LIMA and FP7-ICT-2009-4-248855-N4E). A.M. acknowledges finantial supportthrough FPU from the Spanish ME. P.N. acknowledges supportfrom the Center for Solar Photophysics, an Energy FrontierResearch Center funded by the U.S. Department of Energy andthe Robert A. Welch Foundation C-1222.

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Supporting information for: Quantum Plexcitonics:

A New Approach to Strongly Interacting Plasmons

and Excitons

A. Manjavacas,∗,† F. J. García de Abajo,† and P. Nordlander‡

Instituto de Óptica - CSIC, Serrano 121, 28006 Madrid, Spain, and Department of Physics and

Astronomy, M.S. 61, Rice University, Houston, Texas 77005-1892

E-mail: a.manjavacas@csic.es

w (eV)

50

40

30

20

10

0

2.5 3.0 3.5 4.0 4.5

abso

rptio

n (a

rb. u

nits

)

Classical EM calculationsModel

Figure 1: Comparison of the optical absorption spectrum of the dimer-quantum-emitter systemobtained with classical EM theory (dashed line), and the model developed using the Zubarev’sGreen function method [see Eq. (19) of the main paper] (solid line). We take the emitter excitationenergy εc = 3.35 eV, while the rest of the model parameters are given in Figure 1(b) of the mainpaper.

∗To whom correspondence should be addressed†Instituto de Óptica - CSIC, Serrano 121, 28006 Madrid, Spain‡Department of Physics and Astronomy, M.S. 61, Rice University, Houston, Texas 77005-1892

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Derivation of the optical absorption spectrum in terms of Zubarev’s

Green functions

Within the approximation of linear response, the optical absorption cross section is given by

Fermi’s Golden rule according to

σ(ω) ∝ ∑f

∣∣〈 f ;n−1|H ′ |i;n〉∣∣2 δ (ε f − h̄ω), (1)

where |i〉 represents the initial ground state of the system under consideration; | f 〉 is the corre-

sponding final state, separated from the former by an energy difference ε f ; n is the number of

external photons with frequency ω; and H ′ is the hamiltonian that couples the system to the

external photon field, which is given by the expression

H ′∝ Aa++A+a. (2)

Here, a and A (a+ and A+) are annihilation (creation) operators for the external photons and the ex-

citations of the system, respectively, and therefore, A connects the initial and final states, governing

the optical absorption properties.

Using the hamiltonian of Eq. (2) and the identity

δ (x) =1π

Im{

1x− i0+

}, (3)

we can recast Eq. (1) into

σ(ω) ∝ Im

{∑

f

〈i;n|H ′ | f ;n−1〉〈 f ;n−1|H ′ |i;n〉ε f − h̄ω− i0+

}. (4)

Then, taking into account the action of the photon annihilation operator (a |n〉 =√

n |n−1〉) and

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the orthogonality of the photonic states (〈n| |n′〉= δn,n′), we obtain

σ(ω) ∝ Im

{∑

f

〈i|A | f 〉〈 f |A+ |i〉ε f − h̄ω− i0+

}. (5)

At this point, we assume that A+ connects the initial ground state with a set of final states that

present a common energy ε f . This is a good approximation for the systems considered in this

paper, in which the external photons couple through excitation of either a single particle plasmon

or two degenerate plasmons in a dimer. Therefore, we can extract the denominator of Eq. (5) from

the sum over final states, and using the closure relation for final states ∑ f | f 〉〈 f |= I (within the ε f

reachable-energy shell), the optical absorption spectrum reduces to

σ(ω) ∝ Im{〈i|AA+ |i〉

ε f − h̄ω− i0+

}, (6)

where 〈i|AA+ |i〉 is the expectation value of AA+ in the initial ground state.

We now consider the definition of the retarded Zubarev Green function for the operators that

annihilate and create a system excitation [see Eq. (1) of the main paper]:

〈〈A;A+〉〉ω+i0+ = − ih̄

∫∞

0dtei(ω+i0+)t [⟨A(t)A+(0)

⟩−η

⟨A+(0)A(t)

⟩], (7)

where we have switched to the Heisemberg picture, and therefore, the free evolution of the operator

A is given by the Heisemberg equation of motion,1 which for fixed excitation energy ε f leads to

A(t) = A(0)e−iε f t/h̄. (8)

This is consistent with the noted assumption of a single final-state energy ε f . Eq. (8) allows us to

solve the integral of Eq. (7). We find

〈〈A;A+〉〉ω+i0+ =− 〈A(0)A+(0)〉

ε f − h̄ω− i0+, (9)

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where we have taken into account that the second term of Eq. (7) vanishes when the system is in

the ground state. Finally, noticing that 〈i|AA+ |i〉 and 〈A(0)A+(0)〉 are both the same magnitude in

the Schrödinger and the Heinsemberg picture, respectively, we can substitute Eq. (9) into Eq. (6)

to obtain the desired expression [see Eq. (2) of the main paper].

Derivation of the equation of motion for the retarded Zubarev’s

Green functions

The retarded Zubarev’s Green function in the time domain can be obtained from the Fourier

transform of Eq. (1) of the main paper, as

〈〈A(t);B(0)〉〉=− ih̄

θ(t)⟨[A(t),B(0)]

η

⟩. (10)

Differentiating with respect to the time, we find

iddt〈〈A(t);B(0)〉〉= 1

h̄dθ(t)

dt

⟨[A(t),B(0)]

η

⟩+ 〈〈idA(t)

dt;B(0)〉〉, (11)

which can be simplified using the Heisenberg equation of motion1 to yield

ih̄ddt〈〈A(t);B(0)〉〉= δ (t)

⟨[A(t),B(0)]

η

⟩+ 〈〈[A(t),H ];B(0)〉〉. (12)

Here, H is the hamiltonian of the system. Finally, taking the Fourier transform of Eq. (12), we

readily obtain the desired equation of motion for for the retarded Zubarev’s Green functions [see

Eq. (3) of the main paper].

References

(1) A. Messiah, Quantum Mechanics (Dover, New York, 1999)

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