LETTER pubs.acs.org/NanoLett
Quantum Plexcitonics: Strongly Interacting Plasmons and Excitons A. 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 ABSTRACT: We present a fully quantum mechanical approach to describe the coupling between plasmons and excitonic systems such as molecules or quantum dots. The formalism relies on Zubarev’s Green functions, which allow us to go beyond the perturbative regime within the internal evolution of a plasmonic nanostructure and to fully account for quantum aspects of the optical response and Fano resonances in plasmonexcition (plexcitonic) systems. We illustrate this method with two examples consisting of an exciton-supporting quantum emitter placed either in the vicinity of a single metal nanoparticle or in the gap of a nanoparticle dimer. The optical absorption of the combined emitterdimer structure is shown to undergo dramatic changes when the emitter excitation level is tuned across the gap-plasmon resonance. Our work opens a new avenue to deal with strongly interacting plasmonexcition hybrid systems. KEYWORDS: Quantum optics, plasmon, exciton, plexciton, fano resonance
T
he study and design of devices capable of controlling lightmatter interaction at the nanoscale have been subjects of intense activity over the past decade.1 Metallic nanostructures are ideally suited to this end due to their ability to focus and trap optical energy into subwavelength spatial regions.2 The large fields and the high confinement associated to the plasmonic resonances supported by these systems3 enable strong interactions with other photonic elements such as quantum emitters.4,5 Thanks to the advance in nanofabrication techniques, we are currently approaching length scales in which the quantum behavior of these structures becomes important, thus opening wide horizons for new designs and applications in the novel area of quantum plasmonics.6 Understanding the physics underlying the internal interaction in these systems is important to take advantage of their quantum features, such as collective and single-particle excitations,7 and quantum correlations and interferences.8 The proper theoretical characterization of these processes thus requires a fully quantum mechanical framework. Numerical simulations of Maxwell’s equations, which have been successfully used to model the optical response of a vast number of systems involving metallic nanostructures,9 cannot describe completely the interference and couplings between quasi-particles such as plasmons and excitons. It is therefore important to develop new theoretical tools that overcome such limitations. In this direction, some preliminary results have been recently reported based upon the density matrix formalism.10,11 In this paper, we show that the Zubarev’s Green functions12 method provide a convenient approach for modeling the optical response of plasmons interacting with quantum emitters. This powerful approach enables us to describe the internal evolution of such quantum systems beyond the perturbative regime. r 2011 American Chemical Society
The major advantage of this approach is that it can be systematically extended to include a range of increasingly complex phenomena. For instance, by directly including the coupling between the plasmons and a continuum of electronic states, Fano resonances emerge in a natural way.13,14 The application of this formalism is illustrated with two different examples involving hybrid plexcitons (i.e., systems formed by the interactions of plasmonic nanoparticles with nearby excitonic systems such as quantum dots or molecules). In a first instance, we consider a simple situation in which a quantum emitter is placed close to the surface of a metallic nanoparticle. In a second example, we describe a quantum emitter placed at the gap of a nanoparticle dimer.15 This type of structure is currently attracting much attention16,17 due to its potential application as a platform for quantum information devices. Zubarev’s Green Functions and Their Application to Nano-Optics. The method of Zubarev’s Green functions12 has been successfully applied to different problems in statistical physics and linear response theory.1820 Here, we adapt it to study the optical absorption properties of hybrid systems formed by plasmonic structures and quantum emitters (e.g., molecules or quantum dots). In particular, we compute absorption spectra obtained from the retarded Zubarev Green function of the quantum operators that mediate the photon absorption process. The time-domain Green function of two of these operators contains the information on the evolution of the quantum process that they represent, and its Fourier transform is therefore directly related to the absorption spectrum. Received: February 18, 2011 Revised: April 18, 2011 Published: May 02, 2011 2318
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The retarded Zubarev Green function of two operators A and B is defined in the frequency domain as Z i ¥ iðω þ i0þ Þt dte θðtÞƽAðtÞ, Bð0Þη æ ÆÆA; Bææω þ i0þ ¼ p ¥ ð1Þ where A(t) is the Heisenberg representation21 of operator A, θ(t) is the Heaviside step function, and the brackets [A,B]η = AB ηBA stand for the commutator of bosonic operators (η = 1) or the anticommutator of fermionic operators (η = 1). Notice that an infinitesimally small positive imaginary part is added to the 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; Aþ ææω þ i0þ g
ð2Þ
(a derivation of this expression is given in the Supporting Information), where A (Aþ) is the annihilation (creation) operator of an excitation in the system, resulting from the emission (absorption) of one photon. This type of Green function is commonly calculated by writing its equation of motion pωÆÆA; Aþ ææ ¼ ƽA, Aþ η æ þ ÆƽA, H ; Aþ ææ
ð3Þ
(see the Supporting Information for a derivation of this equation), which depends on another Green function ÆÆ[A, H];Aþææ , where H is the Hamiltonian. In a similar way, this new Green function can be also calculated by writing down its equation of motion. Iterating this process, one obtains a hierarchy of equations that may need to be truncated at some point by applying a physical approximation. Finally, a linear system of equations is derived, from which ÆÆA;Aþææ, and therefore the optical absorption spectrum, is obtained. A Quantum Emitter Interacting with a Metal Nanoparticle. We consider a simple but illustrative system composed of a small metallic nanoparticle with a diameter of tens of nanometers and a quantum emitter (e.g., a quantum dot22,23 or a conjugated polymer molecule24) placed in the vicinity of the nanoparticle. Such a coupled plasmon-exciton system will result in hybrid plasmonic excitonic modes also referred to as Plexcitons.25 We assume that the nanoparticle can support a well-defined bosonic dipolar plasmon mode and neglect higher-multipole modes. This is a reasonable approximation for a noble-metal particle with a radius much smaller than the wavelength. Additionally, we describe the quantum emitter as a fermionic system with only two possible states (ground and excited). The Hamiltonian describing the noninteracting evolution of the fermion and the plasmon is H
0
¼ εd dþ d þ εc cþ c
int
¼ Δdc ½dþ c þ cþ d
Z
ð5Þ
where 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 the excitations of the system, which produce finite widths in the corresponding spectral resonances. Generally, such widths are
Z dω½vd ðωÞfd ðωÞdþ þ vd ðωÞfdþ ðωÞd dω½vc ðωÞfc ðωÞcþ
þ vc ðωÞfcþ ðωÞc
ð6Þ
where fd(ω) and fc(ω) are the annihilation operators of the continuum 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 of the plasmon and the quantum emitter. The total Hamiltonian of our 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 found from the Green function ÆÆd;dþææ [see eq 2]. This should be a good approximation in view of the fact that the absorption cross section of the nanoparticle is generally much larger than that of a molecule or a quantum dot. Using eq 3 and the Hamiltonian derived above, the equation of motion for ÆÆd;dþææ becomes Z ðpω εd ÞÆÆd; dþ ææ ¼ 1 Δdc ÆÆc; dþ ææ dω0 vd ðω0 ÞÆÆfd ðω0 Þ; dþ ææ
ð7Þ From this expression, it is clear that we need to compute two additional Green functions: ÆÆc;dþææ and ÆÆfd(ω);dþææ. The first of them can be easily obtained from its equation of motion ðpω εc ÞÆÆc; dþ ææ ¼ Δdc ÆÆð1 2cþ cÞd; dþ ææ Z dω0 vc ðω0 ÞÆÆð1 2cþ cÞfc ðω0 Þ; dþ ææ
ð8Þ
Therefore, we see that new Green functions emerge that need to be calculated. The iteration of this process would produce an infinite hierarchy of equations of motion, but we truncate it at this point by approximating the operator cþc by its expectation value Æcþcæ = nc. The presence of this term is the result of the fermionic character of the quantum emitter. We still need to deal with Green functions containing information related to the inelastic decay processes (i.e., ÆÆfd(ω);dþææ, and ÆÆfc(ω);dþææ). Upon inspection of the Hamiltonian, we conclude that these Green functions only depend on those derived above, and they are given by
ðpω pω0 ÞÆÆ fd ðω0 Þ; dþ ææ ¼ vd ðω0 ÞÆÆd; dþ ææ
ð4Þ
where d and c (dþand cþ) are the annihilation (creation) operators for the nanoparticle plasmon and the quantum emitter fermion of energies εd and εc, respectively. The plasmon-fermion interaction is modeled by the Hamiltonian H
the result of the inelastic interaction with a continuum of modes. For instance, a plasmon can decay radiatively by emitting one photon and nonradiatively through the generation of electronhole pairs, phonons, etc. We describe these inelastic interactions by adding a term to the Hamiltonian Z Z þ dωpωfcþ ðωÞfc ðωÞ H decay ¼ dωpωfd ðωÞfd ðωÞ þ
ðpω pω0 ÞÆÆ fc ðω0 Þ; dþ ææ ¼ vc ðω0 ÞÆÆc; dþ ææ
ð9Þ
Substituting the first of these expressions into eq 7 and introducing the positive infinitesimal imaginary part of the frequency that appears in the definition of Zubarev’s Green function [see eq 1], we find the integral Z Z 0 2 jvd ðω0 Þj2 0 jvd ðω Þj þ iπjvd ðωÞj2 ¼ P dω dω0 0 pω pω i0þ pω0 pω Γd ð10Þ ¼ δωd þ i 2 2319
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which we have evaluated using the SokhatskyWeierstrass theorem. Here, P denotes the Cauchy principal value, δωd represents a frequency shift, and Γd = 2 π|vd(ω)|2 is the inelastic decay rate of the dipolar plasmons. Dealing in a similar way with the remaining fermionic decay channels, we can transform eqs 7 and 8 into Γd pω εd þ δωd þ i ÆÆd; dþ ææ ¼ 1 Δdc ÆÆc; dþ ææ ð11Þ 2 and
Γc ÆÆc; dþ ææ ¼ Δdc ð1 2nc ÞÆÆd; dþ ææ pω εc þ ð1 2nc Þ δωc þ i 2
ð12Þ which can be easily solved to obtain the desired Green function ÆÆd; dþææ. Finally, inserting this result into eq 2, we obtain the optical absorption spectrum of the plasmon-fermion system 91 8 > > > > = < Γd Δ2dc ð1 2nc Þ σ Im pω εd þ δωd þ i > Γc > 2 > pω εc þ ð1 2nc Þ δωc þ i > ; : 2
ð13Þ This equation clearly shows that the plasmonic resonance is modified by the interaction with the quantum emitter. However, although the fields near a metallic nanoparticle can be large due plasmonic enhancements, any realistic value of the coupling constant Δdc of this system is too small to induce an observable change of the spectrum. A noticeable effect can be nonetheless observed when the emitter is situated in a hot spot of a more complex plasmonic structure, such as the gap of a particle dimer in which a well-defined single gap mode can be clearly resolved for small gap distance. We consider this situation later in this paper, but first we describe the more complex interaction between the emitter 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 the quantum emitter and the plasmon is largely enhanced, thus producing noticeable effects in the optical absorption spectrum. More precisely, we consider two identical metallic nanoparticles with diameter of tens of nanometers separated by a gap of a few nanometers. A quantum emitter is placed in this gap, where the field-amplitude enhancement due to the plasmonic resonances of the particles can easily reach 2 orders of magnitude with respect to the value of an external plane wave.3 The optical response of this system has been recently studied using classical electrodynamics methods.16,17 We assume each particle to be supporting just one dipole plasmon and one quadrupole plasmon with m = 0 azimuthal symmetry with respect to the dimer axis. Only modes with this symmetry can be excited because we consider the external field to be oriented along the dimer axis. In a real system, higher-order multipoles are also contributing for small separations, although the dipoles are dominant for the separations discussed below. In fact, the quadrupole can be considered to effectively account for the effects of all higher-order modes combined. The quantum emitter is described as a two level system like in the previous section. The Hamiltonian that determines the behavior of this nanostructure contains again three terms H = H0 þ Hint þ Hdecay. The noninteracting part reads H
0
þ þ ¼ εd ½d1þ d1 þ d2þ d2 þ εq ½qþ 1 q 1 þ q 2 q 2 þ εc c c
ð14Þ
where di and qi are the dipolar and quadrupolar plasmon bosonic annihilation operators of particle i = 1,2, respectively. Furthermore, εq is the quadrupolar plasmon energy, while the rest of the operators have a similar meaning as in eq 4. The interaction Hamiltonian Hint must describe all possible couplings of the different system excitations. In our particular case, the dipolar and quadrupolar plasmons of one particle can interact with the excitations of the other one via dipoledipole (Δdd), dipolequadrupole (Δdq), and quadrupolequadrupole (Δqq) coupling constants. In addition, the quantum emitter interacts with both the dipole plasmons (Δdc) and quadrupole plasmons (Δqc). Therefore, we have H
int
þ þ þ ¼ Δdd ½d1þ d2 þ d2þ d1 Δqq ½qþ 1 q2 þ q2 q1 Δdq ½d1 q2 þ q2 d1 þ þ þ þ Δdq ½d2þ q1 þ qþ 1 d2 Δdc ½d1 c þ c d1 Δdc ½d2 c þ c d2
þ þ þ Δqc ½qþ 1 c þ c q1 Δqc ½q2 c þ c q2
ð15Þ
Finally, the decay part of the Hamiltonian is given by eq 6, where we now need to sum over both dipole plasmons and include extra terms for the quadrupole plasmons. For particles with diameters of tens of nanometers, the quadrupoles couple only weakly to the light. As in the previous section, we can neglect the direct coupling of the emitter to incident light. We therefore assume that only the dipolar plasmons couple efficiently to the external photons. (However, the extension of our calculations to include the direct coupling to the emitter and the quadrupoles is straightforward.) With this approximation, we just need to obtain the Green function ÆÆD; Dþææ, where D is a linear combination of the dipolar plasmon annihilation operators of particles 1 and 2 compatible with the symmetry of the external field. We choose D = d1 þ d2, which implies that the incident photons are absorbed at the two particles with the same phase. This represents the situation in which the external field is incident perpendicularly to the dimer axis, and therefore, only the bonding mode15 of the dimer is excited. Using eq 3 and the Hamiltonian derived above, the equation of motion of the Green function ÆÆD;Dþææ reads ðpωεd þ Δdd ÞÆÆD; Dþ ææ ¼2 Δdq ÆÆQ ; Dþ ææ 2Δdc ÆÆc; Dþ ææ Z dω0 vd ðω0 ÞÆÆ f D ðω0 Þ; Dþ ææ ð16Þ where Q = q1 þ q2 and fD = fd1 þ fd2. Like in the previous section, we are encountering additional Green functions. In particular ÆÆQ;Dþææ, whose equation of motion is ðpωεq þ Δqq ÞÆÆQ ; Dþ ææ ¼ Δdq ÆÆD; Dþ ææ 2Δqc ÆÆc; Dþ ææ Z
dω0 vq ðω0 ÞÆÆ fQ ðω0 Þ; Dþ ææ
ð17Þ
with fQ = fq1 þ fq2. Similarly, the dynamics of ÆÆc;Dþææ is governed by ðpωεc ÞÆÆc; Dþ ææ ¼ Δdc ð1 2nc ÞÆÆD; Dþ ææ Δqc ð1 2nc ÞÆÆQ ; Dþ ææ Z
ð1 2nc Þ
dω0 vc ðω0 ÞÆÆ fc ðω0 Þ; Dþ ææ
ð18Þ
where we have truncated the hierarchy of equations of motion by replacing the operator cþc in the Green functions by its expectation value nc. We also need to follow three additional Green functions involving the operators fD, fQ, and fc, which appear in the equations of motion derived above. Examining the Hamiltonian, it is clear that they are given by expressions similar to eq 9, 2320
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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 the system under study. (b) Values of the different model parameters (see text). (c) Energy diagrams of the quantum emitter (fermionic character), and the dipolar and quadrupolar plasmons supported by the nanoparticles (bosonic character). (d) Optical absorption spectrum of the dimerquantumemitter system, computed for different values of the emitter resonance energy. A Fano resonance is clearly observed as a result of the coupling between the emitter and the particle plasmons.
which can be immediately inserted into eqs 1618. Then, following the same procedure as in the previous section to deal with the integrals over frequency [see eq 10], we end up with a linear system of three equations and three unknowns: ÆÆD;Dþææ, ÆÆQ;Dþææ, and ÆÆc;Dþææ. Finally, solving this system of equations we obtain ÆÆD,Dþææ, and therefore, via eq 2, the optical absorption spectrum 8 > > > > >
> > > > :
" 2Δ2dc ð1 2nc Þ Sc
#2 91 2Δqc Δdc ð1 2nc Þ > > > > Δdq > = Sc Sq
2Δ2qc ð1 2nc Þ Sc
> > > > > ;
ð19Þ where we have defined Sd = pωεd þ Δdd þ δωd þ iΓd/2, Sq = pωεq þ Δqq þ δωq þ iΓq/2, and Sc = pωεc þ (12nc)[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 parameters involved in eq 19. We illustrate this by considering two silver nanoparticles of 10 nm radius separated by a gap of 4 nm. We take the system to be initially prepared in the ground state, so that nc = 0 for the emitter placed in the dimer gap. Moreover, the frequency shifts δωd and δωq can be accounted for by renormalizing the energies of the corresponding states εd, and εq, respectively. Notice that we keep δωc as it cannot be reabsorbed when nc 6¼ 0. We extract the energies and widths of the dipolar and quadrupolar plasmons from classical electromagnetic simulations of the absorption cross section for a single silver nanoparticle using Mie theory. In a similar way, we obtain the coupling constants describing the interaction between the dipolar and quadrupolar plasmons by fitting eq 19 with Δdc = Δqc = 0 (i.e., without emitter) to an electromagnetic simulation of the dimer absorption cross section using a multiple-scattering approach.27 The resulting values for the different parameters are given in Figure 1b. Figure 1 shows the optical absorption spectrum of the system under study, computed from eq 19, for different energies of the quantum emitter resonance. The width of this resonance has been taken to be Γc = 4 meV, which is a realistic value for actual
quantum dots.28 It is important to note that we have considered quantum emitters of resonance energy lying close to the dimer plasmons. For a given emitter energy within the 33.5 eV range, the dimer plasmon frequencies can be easily tuned by selecting the size, shape, and separation of the nanoparticles29 (e.g., using nanoshells of appropriate thickness30). We have calculated the strength of the coupling between the quantum emitter and the dipolar and quadrupolar plasmons, and the value of δωc by fitting eq 19 to a simulation performed with classical electromagnetic theory, as shown in the Supporting Information. In this model the quantum emitter is represented by a point dipole with a linear polarizability31 R(ω) = μ2/(εc pω iΓc/2), where μ is the transition dipole. In our particular example we assume μ/e = 0.3 nm, which is a realistic value for a quantum dot.32 Using this transition dipole we obtain the values for the different model parameters shown in Figure 1b. Thanks to the large plasmonic fields existing in the dimer gap, this system exhibits large values of the coupling constants Δdc and Δqc as compared 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 quadrupolar dimer modes around 3.3 (≈ εd Δdd) and 3.5 eV (≈ εq Δqq); and the excitonic mode, which is clearly visible as a Fano resonance with a line shape that strongly depends on the energy εc. The Fano resonance results from the interaction between a continuum of modes and a narrow discrete mode.14 In our example, the quantum emitter resonance is the narrow mode, while the dipolar and quadrupolar plasmons, with their larger widths, play the role of a continuum. As expected these resonances can be fitted approximately using the conventional Fano line shape with |q| , 1. For instance, in the case of the curve of Figure 2b corresponding to εc = 3.35 eV we 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 to observe Fano resonances in this type of system. Figure 2 also shows that the Fano resonance displayed for small values of Γc disappears as this parameter approaches the value of the dipolar plasmon width Γd. In Figure 3, we illustrate the critical role played by the strength of the coupling between the quantum emitter and the dipolar plasmon Δdc in determining the shape of the optical absorption spectrum. For small values of Δdc, the dip associated to the Fano 2321
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Figure 2. Dependence of the optical absorption spectrum on the quantum emitter resonance width Γc. We take the emitter excitation energy εc = 3.35 eV, while the rest of the model parameters are given in Figure 1b. The Fano resonance disappears as Γc increases.
Figure 4. Dependence of the optical absorption spectrum on the expectation value of the emitter excited level nc for different values of the emitter excitation energy. The rest of the model parameters are given in Figure 1b. The absorption spectrum changes dramatically as nc changes from 0 to 1.
Figure 3. Dependence of the optical absorption spectrum on the strength of the coupling between the dipole plasmons and the quantum emitter Δdc. We take the emitter excitation energy εc = 3.35 eV, while the rest of the model parameters are given in Figure 1b. A dramatic dependence of the spectral shape on Δdc is observed.
resonance almost disappears. For increasing Δdc, the dip becomes deeper and the separation between the two resulting peaks becomes more pronounced. Actually, when this coupling constant is larger than the dipolar plasmon half width Γd/2 we can alternatively interpret the resulting line shape as vacuum Rabi splitting.16 Nonlinear Effects and Beyond. It is important to stress that our results reproduce the simulations performed in previous works.16,17 However, our methodology allows us to go one step forward in the understanding of the physical mechanisms governing the response of the system. First of all, we can clearly identify the origin of the peaks and other features that characterize the investigated optical absorption spectra. Furthermore, our model can capture the nonclassical behavior of the different elements of the system. For instance, in a situation in which the quantum emitter is not initially prepared in its ground state, and hence, the expectation value nc does not vanish. In such case, the optical response of the system changes dramatically [see eq 19]. In particular, under strong optical pumping this expectation value should be ∼1/2, for which the interaction between the quantum emitter and the dipolar plasmon disappear, leading to an inhibition of the Fano resonance. Such nonlinear Fano resonances have been observed in quantum-well structures and are of clear fundamental importance.4,33 This can be clearly observed in Figure 4, which shows the dependence of the optical absorption
spectrum on the expectation value nc, for different emitter excitation energies εc. As nc grows from 0 to 0.5 the Fano resonance becomes weaker and finally disappears, reemerging again for higher values. At the same time, the position of the Fano resonance is shifted toward higher energies. This behavior, which is associated to the saturation of the quantum emitter, emerges from the fermionic nature of this system. For this reason, although it could be modeled a posteriori using a classical or a semiclassical approach by assuming a change in the oscillator strength, it could not be predicted a priori, unless a fully quantum approach is employed. In practical terms, saturation can be obtained with a pumping-light frequency differing from the probing frequency, but separated from it by less than the width of the emitter excitation. The nonlinear response associated to this nonclassical behavior opens the possibility of externally controlling the optical absorption spectrum of the system. The Zubarev’s Green functions12 method presented in this paper is conceptually much simpler than standard Green’s function or density matrix approaches since the only formalism that is required is the evaluation of simple commutation relations between operators. However, the major advantage of the present approach is that all dynamics are obtained nonperturbatively directly from the Hamiltonian. Because of this, by simply adding more terms to the Hamiltonian, it is straightforward to include more complex interactions such as the coupling to the continuum which introduced the broadening of the plasmon and exciton states already discussed. Most importantly, our method provides a simple approach for including much more complicated effects such as strong correlation, coupling between two emitters, coupling to phonons and other collective excitations. For instance, an interesting scenario is presented when two quantum emitters are placed at the gap of a particle dimer. The interaction between emitters can be affected by Coulomb 2322
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Nano Letters repulsion, which can be incorporated into our formalism by adding a Hubbard term to the Hamiltonian as it is commonly done in the Anderson model.34 Such a term prevents the two quantum emitters from simultaneously lying in excited states and may therefore introduce additional useful tunable nonlinear behavior of relevance for plexciton-based quantum information devices. Conclusions. In summary, we have presented a new approach based on the Zubarev’s Green functions method for the study of the optical properties of plasmonic systems interacting with quantum emitters (e.g., molecules or quantum dots). The formalism describes the quantum internal evolution of such fermionic-bosonic hybrid systems beyond the perturbative regime. Its power is here demonstrated with two illustrative examples: (1) a system consisting of a quantum emitter placed close to a metallic nanoparticle and (2) a quantum emitter placed in the gap of a nanoparticle dimer. Using realistic parameters, we show that the optical absorption spectrum of the dimer-quantum-emitter system can exhibit Fano resonances resulting from the interaction between the quantum emitter and the gap plasmon, in agreement with previous work.16,17 Our approach can be straightforwardly generalized to include more complicated interactions. The formalism is likely to become a powerful tool for the description of plasmonexcition interactions in plasmonic transistors, modulators, and quantum information devices.
’ ASSOCIATED CONTENT
bS
Supporting Information. An extra figure related to the determination of the model parameters and the derivations of eq 2 and eq 3 are provided. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This work has been supported by the Spanish MICINN (MAT2010-14885 and Consolider NanoLight.es) and the European Commission (FP7-ICT-2009-4-248909-LIMA and FP7ICT-2009-4-248855-N4E). A.M. acknowledges finantial support through FPU from the Spanish ME. P.N. acknowledges support from the Center for Solar Photophysics, an Energy Frontier Research Center funded by the U.S. Department of Energy and the Robert A. Welch Foundation C-1222.
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dx.doi.org/10.1021/nl200579f |Nano Lett. 2011, 11, 2318–2323
