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1.INTRODUCTIONStrong coupling of localised surface plasmons in metal-dielectric structures with excitons in dye molecules or semiconductors attracts intense interest due to emerging applications such as ultrafast reversible switching,1–3 quantum computing,4, 5 and light harvesting.6 An extremely strong field confinement on the length scale well below the diffraction limit provided by plasmonic structures gives rise to exciton-plasmon coupling that is much stronger than the coupling to cavity modes in semiconductor microcavites. Strong coupling regime between two systems is established when the energy exchange between them takes place faster than the energy dissipation, which leads to an anticrossing gap (Rabi splitting) in the dispersion of mixed states.7 While relatively weak Rabi splittings ~ 1 meV were reported for semiconductor quantum dots (QDs) coupled to a cavity mode,8–10 much greater splittings (up to 500 meV) were observed for surface plasmons coupled to excitons in J-aggregates,11–20 various dye molecules,21–25 semiconductors QDs,26–28 or two-dimensional atomic crystalls.32–35 In the classical picture of coupled oscillators,7 the interaction between a quantum emitter (QE), modeled here by a two-level system with frequency ω0, and a cavity (or plasmon) mode with frequency wm, gives rise to mixed states that show up in optical spectra through splitting of scattering or emission peaks into two polaritonic bands separated (for ωm = ω0) by Here μ is the QE dipole matrix element and If an ensemble of N QEs is coupled to a cavity mode, they form a collective state comprised of all QEs oscillating in sync with each other.40 For such a collective state, the coupling scales as This picture is affected dramatically if an ensemble of N QEs is placed near a plasmonic nanostructure, which is normally characterized by strongly varying local fields. For example, in a typical experimental setup, the QEs (excitons in J-aggregates or QDs) are embedded in a dielectric shell enclosing a metal core supporting localized surface plasmon (see Fig. 1). Strong exciton-plasmon coupling requires large plasmon local density of states (LDOS) facilitating an efficient energy transfer (ET) between a plasmon and a QE placed in a strong local field region. In plasmonic structures, the local fields can be very strong close to metal surface, especially near sharp features such as narrow tips or surface irregularities, but fall off rapidly away from the surface. In this case, the coupling between individual QEs and resonant plasmon mode can vary in a wide range, so that the classical “giant oscillator” picture is no longer useful and, instead, one has to resort to the underlying mechanism of energy exchange between the plasmon and QEs. Furthermore, since for sufficiently remote QEs, the individual QE-plasmon ET rates are small, it is evident that the ensemble QE-plasmon coupling should saturate as the region QEs are distributed in expands. Saturation of Rabi splitting with an increasing system size was recently reported for molecular excitons in J-aggregates embedded in dielectric shell enclosing a gold nanoprism.20 Figure 1.Schematics of exciton-plasmon coupling mediated by cooperative energy transfer between a collective state and resonant plasmon mode. (b) Schematics of open plasmonic system with QEs embedded in dielectric shell enclosing a metallic core. ![]() In this Letter, we develop a model for exciton-pasmon coupling based upon microscopic picture of energy exchange between system components. We establish an explicit relation between the ensemble QE-plasmon coupling and the rate of cooperative energy transfer (CET) between a collective state and resonant plasmon mode (see Fig. 1), and estimate the energy-exchange frequency in the strong coupling regime. By defining carefully the plasmon mode volume in terms of plasmon LDOS, we show that, for a single QE at a hot spot near sharp metal tip, the QE-plasmon coupling scales as where 2.PLASMON LDOS, MODE VOLUME AND ENERGY TRANSFER RATETo establish a relation between cavity-like coupling (1) and microscopic picture of energy exchange in plasmonic systems, we employ a classical approach for plasmons and describe excitons by two-level systems, reffered to as QEs. We assume that QEs are placed near a metal-dielectric structure with characteristic size smaller than the radiation wavelength described by a complex dielectric function where Here, E(r) is the plasmon field, which we chose to be real, satisfying the Gauss’s law, while the factor ωm/4Um, is the plasmon pole residue in the complex frequency plane.42 Using Eqs. (3) and (4), we obtain the projected plasmon LDOS as a Lorentzian centered at the plasmon frequency, where Q = ωm/γm, is the plasmon quality factor. The plasmon LDOS describes the distribution of plasmon states in unit volume and frequency interval. Accordingly, its frequency integral, where In order to relate At resonance frequency where To illustrate ET rate’s sensitivity to the QE position and system geometry, in Fig. 2 we show Figure 2.Normalized QE-plasmon ET rate is plotted against the QE distance from Au nanorod tip for several nanorod aspect ratios. Inset. Schematics of a QE placed at a distance d from a tip of Au nanorod in water. ![]() Let us now turn to an ensemble of N QEs with dipole moments pi = μηi situated at positions ri near a plasmonic structure. Each QE interacts, via the coupling The diagonal elements of plasmon coupling matrix Eq. (10) are complex, The off-diagonal elements of plasmon coupling matrix Eq. (10) describe plasmonic correlations between QEs which give rise to the collective states. These states are represented by the eigenstates ψ of the coupling matrix (10) satisfying where Since the imaginary part of the eigenvalue λΝ saturates the ET rates from the QEs to resonant plasmon mode, the rest of the collective states are not coupled to that mode but, in principle, can still couple to the off-resonant modes and radiation field. Note, however, that for large ensembles, coupling of collective states to the off-resonant modes is relatively weak,54,55 while large plasmonic Purcell factors ensure that coupling to the radiation field is relatively weak as well, implying that the exciton-plasmon energy exchange at the CET rate (11) is the dominant energy flow channel in the system. We stress that, in plasmonic systems, the collective states are formed due to QEs’ correlations via the local fields that can vary strongly near a plasmonic structure. Therefore, these states are distinct from the superradiant and subradiant states, which emerge due to QEs’ coupling to the common radiation field that is nearly uniform on the system scale. 3.EXCITON-PLASMON COUPLING AND ENERGY EXCHANGEConsider first a single QE situated near a plasmonic resonator with its frequency ω0 close to the plasmon frequency ωm. We consider systems with characteristic size smaller the radiation wavelength and, therefore, consider only the near-field coupling between a QE and resonant plasmon mode; note, however, that for larger systems, radiative (superradiant) coupling between them can significatly affect the optical spectra in the strong coupling regime.18 Using Eq. (8), we can now relate the QE-plasmon coupling (1) to the QE-plasmon ET rate as The above expression for g implies a relation between the QE-plasmon coupling and the Purcell factor: g2 = To describe the transition to strong coupling regime, we note that, within classical model of coupled oscillators, the spectrum splits into upper and lower polaritonic bands with complex frequencies7 where we used the relation (12). For In the strong coupling regime, both polariton bands are characterized by constant spectral width γm/2, while their central frequencies are separated by the Rabi splitting: Thus, in terms of QE-plasmon ET rate, the transition to strong coupling regime take place at Note that the polaritonic bands become spectrally distinct when Rabi splitting exceeds their linewidth γm/2, which corresponds to the condition The onset condition (15) implies that, for plasmonic resonators with high quality factor and damping rate γm/2, which is similar to that for optical field intensity in the strong coupling regime [see Eq. (14)]. In this model, the transition from strong to weak coupling regime is analogous to the transition from underdamped to overdamped regime taking place at Let us now turn to an ensemble of N QEs near a plasmonic structure. As discussed in the previous section, plasmonic correlations between the QEs give rise to a collective state that exchanges its energy with resonant plasmon mode at the CET rate where The ensemble QE-plasmon coupling (17) is related as The ensemble QE-plasmon coupling gN is related to individual QE couplings In the above analysis, we assumed same excitation frequencies for all QEs in the ensemble. In the experiment, however, the QE frequencies are distributed within some interval due to, e.g., direct dipole-dipole interactions between the QEs or, in the case of semiconductor QDs, their size variations. Here we note that the ET rate between a donor and an acceptor is determined by the spectral overlap of their respective emission and absorption bands.7 Therefore, as long as the ensemble inhomogeneous broadening stays within the broad plasmon resonance band, the QE-plasmon energy exchange mechanism remains largely unaffected. 4.EXCITON-PLASMON COUPLING NEAR SHARP METAL TIPThe largest values of exciton-plasmon coupling are achieved for QEs placed in a region with large plasmon LDOS that provides an efficient ET between a QE and a plasmon mode. Here, we consider a single QE situated near a sharp tip of a small plasmonic structure, such as a metal nanorod, where the field confinement can be extremely strong (hot spot). To estimate the QE-plasmon coupling, we note that the Gauss’s law implies where we assumed that only in the metallic region the dielectric function is dispersive. The integral over metallic region in Eq. (19) depends on the characteristic size of that region lm relative to the skin penetration length ls (about 20 nm for Au in plasmon frequency domain). For small structures with lm < ls, the QE-plasmon coupling scales as To elucidate the relative importance of these two enhancement sources, below we estimate the QE’s coupling g to a plasmon mode oscillating along the tip of a small metal structure (see Fig. 3). For small systems, the plasmon field does not significantly change inside the metallic structure while falling off rapidly outside of it, so the largest field enhancement takes place near the tip. For QE polarized along the tip (see Fig. 3), the coupling (19) at the system symmetry axis (z-axis) can be estimated as Figure 3.Normalized QE-plasmon coupling is plotted against the QE distance from Au nanorod tip for several nanorod aspect ratios. Inset. Schematics of a QE placed at a distance d from a tip of Au nanorod in water. ![]() where subscripts in and out refer, respectively, to the fields at the interface on metal and dielectric sides. Using the boundary condition where The maximal value of gtip is achieved in a close proximity to the tip, where indicating that the plasmonic enhancement is due to both the geometric volume effect and the plasmon field enhancement characterized by In Fig. 3, we plot distance dependence of the QE-plasmon coupling for a QE near the tip of Au nanorod in water. Note that for a prolate spheroid used here to model the nanorod, the expression (21) is exact. The curves show calculated 5.SATURATION OF EXCITON-PLASMON COUPLING IN OPEN SYSTEMSIn open plasmonic systems, the QEs are distributed within some region outside the metal structure, e.g., within dielectric shell, with dielectric constant εd, enclosing a metallic core [see Fig. (1)]. In such systems, the plasmon field E(r) can vary substantially within the QE region and, in particular, falls off rapidly away from the metal surface. If QEs are uniformly distributed, with concentration n, within some volume V0, the sum in Eq. (18) is replaced by the integral over V0, and the corresponding QE-plasmon coupling g0 takes the form where the factor 1/3 comes from orientational averaging. In contrast to individual QE-plasmon coupling (19), the ensemble coupling (24) is determined by the ratio of integrated field intensities over QE and metallic regions. If the QE region is sufficiently extended, the remote QEs do not interact with the plasmon mode and, therefore, the integral Let us now show that the actual value of saturated coupling gs does not depend directly on the size or shape of the metal structure as well and, remarkably, can be obtained explicitly for any plasmonic system with characteristic size below the diffraction limit. Indeed, due to the Gauss’s law, both volume integrals in Eq. (24) reduce to surface integrals over the system interfaces, including the common metal-dielectric interface. After matching the normal field components at this interface and disregarding, in the saturated case, the outer boundary of the QE region, the ratio of integrated intensities in Eq. (24) is found as Using the relation and the transition onset to strong coupling regime for large ensembles, Note that saturation of the QE-plasmon coupling in the strong coupling regime was recently reported for photolumenescence of molecular excitons in J-aggregates embedded in dielectric shell enclosing an Au nanoprism.20 In Fig. 4, we show the calculated QE-plasmon coupling g0, given by Eq. (24), for QEs distributed uniformly within SiO2 shell enclosing an Au nanorod. This core-shell system is modeled by two confocal spheroids with major semi-axes a and a1 corresponding to the Au/SiO2 and Si2O/H2O interfaces, respectively. With expanding QE region, the coupling g0 saturates to the value gs given by Eq. (25). Saturation is faster for more elongated particles possessing hot spots near their tips, in which case the QE-plasmon coupling reaches 90% of its saturated value for a1/a = 2. Figure 4.Calculated QE-plasmon coupling (24) for QEs uniformly distributed within SiO2 shell enclosing an Au nanorod normalized by saturated coupling (25) is plotted against increasing shell thickness for several aspect ratios. Inset. Schematics of a core-shell Au/SiO2 nanorod in water. ![]() Finally, let us compare the coupling parameters for large QE ensembles coupled to plasmons and to cavity modes in the case when QEs saturate their respective mode volumes. While in plasmonic cavities, the QE-plasmon coupling is strongly enhanced relative to QEs’ coupling to a cavity mode, this is not the case for open plasmonic systems, where only a small fraction of QEs close to metal surface interacts strongly with the plasmon mode. Indeed, let us assume that QEs distributed uniformly, with concentration n, inside a microcavity. The QE-cavity coupling is also given by Eq. (17), but with cavity mode volume Vcav given by9 where Comparing 6.CONCLUSIONSIn conclusion, we developed a model for exciton-plasmon coupling based on a microscopic picture of energy exchange that drives the transition to strong coupling regime. Plasmonic correlations between the QEs give rise to a collective state that transfers its energy cooperatively to a resonant plasmon mode at a rate equal to the sum of individual QE-plasmon ET rates. It is this CET rate that, along with plasmon decay rate, determines the QE-plasmon coupling as well as the energy-exchange frequency in the strong coupling regime. 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