1.

INTRODUCTION

Strong 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 w_m, gives rise to mixed states that show up in optical spectra through splitting of scattering or emission peaks into two polaritonic bands separated (for ω_m = ω₀) by , where γ₀ and γ_m are, respectively, the QE’s and mode’s decay rates, and g is the coupling parameter,⁹

Here μ is the QE dipole matrix element and is the mode volume characterizing field confinement at the QE position. Since typically , the onset of transition to the strong coupling regime is , which demands sufficiently strong field confinement (or small mode volume. For a QD placed inside a microcavity, the cavity mode volume is at least , where λ is the wavelength, resulting in Δ ~ 1 meV.⁹ However, if a QE is located at a “hot spot” near a plasmonic structure characterized by extremely strong field confinement, e.g., near a sharp metal tip or in a gap between two metal structure, the Rabi splitting can reach much greater values even though the plasmon decay rate is much higher than that for cavity modes.^36–39

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 with the ensemble size, while the rest of collective states are “dark”, i.e., they interact only weakly with the electromagnetic field. Importantly, such scaling implies that individual QEs are coupled to the cavity mode with approximately equal strength, so that the mode, in fact, interacts with one “giant oscillator” whose amplitude is enhanced by the factor N relative to individual QE.

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 V_met is the metal volume fraction that confines the plasmon field, and provide an analytical expression for the enhancement factor relative to exciton coupling to a cavity mode. For an ensemble of QEs near a plasmonic structure, we show that the ensemble coupling parameter scales as , where is the average plasmon mode volume in the QE region. If QEs are placed in a region with nearly constant, however large, plasmon LDOS, the coupling exhibits the usual cavity-like scaling . However, in open plasmonic systems, if QEs are uniformly distributed, with concentration n, in an extended region saturating the plasmon mode volume (see Fig. 1), the ensemble QE-plasmon coupling saturates to some value that, for system size below the diffraction limit, has a universal form

where is the metal dielectric function and is the dielectric constant in the QE region. The saturated coupling g_s is independent of system’s size and shape, except indirectly via the plasmon frequency w_m. Surprisingly, g_s is comparable with the coupling between QEs, uniformly distributed inside a microcavity with the same concentration, and a cavity mode, suggesting that large Rabi splittings, observed in open plasmonic systems, are likely due to high QE concentrations.

2.

PLASMON LDOS, MODE VOLUME AND ENERGY TRANSFER RATE

To 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 . The interaction of a QE, located at a position r with dipole orientation n, with eloctromagnetic environmentby is characterized by projected LDOS⁷

where is the rescaled electromagnetic dyadic Green function in the presence of plasmonic structure ( is the standard Green dyadic for Maxwell equations and k is the wave vector). In the near field limit, the Green function splits into free-space and plasmon contributions, , where the plasmon Green function, for frequency ω near the plasmon mode frequency ω_m, has the form:^41,42

Here, E(r) is the plasmon field, which we chose to be real, satisfying the Gauss’s law, , and tensor product of fields is implied. In Eq. (4), U_m is the plasmon mode energy defined in the standard way as ⁴³

while the factor ω_m/4U_m, 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 = ω_m/γ_m, is the plasmon quality factor. The plasmon LDOS describes the distribution of plasmon states in unit volume and frequency interval. Accordingly, its frequency integral, , defines the projected plasmon mode density as,^41,42

where is the projected plasmon mode volume that characterizes the plasmon field confinement at a point r in the direction n. We stress that, for characteristic system size smaller than the radiation wavelength, the plasmon mode volume is a real function of r defined in terms of real plasmon fields at that position.⁴² For larger systems, where plasmon modes are strongly hybridized with the radiation continuum, field confinement is not a well-defined notion but, in this case, the QE-plasmon interaction can still be treated accurately using the expansion over quasi-normal modes characterized by complex mode volumes.^44–48 Within our analytical approach, the above connection between the plasmon mode volume and plasmon LDOS allows relating the QE-plasmon coupling with the rate of energy transfer (ET) between them.

In order to relate to the QE-plasmon ET rate , we note that the latter can be expressed via the plasmon Green function as . Using Eq. (4), we obtain the frequency-dependent ET rate as a Lorentzian centered at the plasmon frequency,

At resonance frequency the QE-plasmon ET rate takes the form

where is given by Eq. (7). Normalizing by the free-space radiative decay rate , we obtain the Purcell factor⁴⁹ for a QE coupled to a plasmonic resonator: .

To illustrate ET rate’s sensitivity to the QE position and system geometry, in Fig. 2 we show near the tip of Au nanorod in water. Nanorod was modeled by a prolate speroid with semimajor and semiminor axes a and b, respectively, and the experimental Au dielectric function was used in all calculations. To highlight the dependence of on system geometry, we normalize it by its value for a sphere with radius a and plot the result against the normalized distance d/a for several aspect ratios a/b. With increasing aspect ratio, i.e., reducing the nanorod volume relative to the sphere, the normalized ET rates increase dramatically near the nanorod tip, which translates to analogous behavior of the QE-plasmon coupling, as we discuss in detail later in the paper.

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 p_i = μη_i situated at positions r_i near a plasmonic structure. Each QE interacts, via the coupling , with the common electric field generated by all QEs, where ) is the near-field electromagnetic Green’s function in the presence of plasmonic structure. Due to electromagnetic correlations between QEs, collective states are formed that are described by the eigenstates of the Green’s function matrix at QEs’ positions projected onto QEs’ dipole moments:⁵⁰ . Note that this coupling matrix includes direct dipole-dipole interactions between the QEs which are causing random shifts of the QEs’ energies. However, for random dipole orientations, the direct dipole coupling vanishes on average,^51,52 while its fluctuations contributes, among other factors, to inhomogeneous broadenning of the QE energies (we will return to this point later). Furthermore, for system size smaller than the radiation wavelength, the radiative (superradiant) coupling between the QEs is relatively weak as well and is not considered here. Therefore, near the plasmon resonance, the coupling matrix is dominated by the plasmon Green function Eq. (4), and we obtain⁵³

The diagonal elements of plasmon coupling matrix Eq. (10) are complex, , and their real and imaginary parts describe, respectively, the frequency shifts δω⁽ⁱ⁾ and decay rates of individual QEs due to interaction with a plasmon mode; the former contribute to inhomogeneous broadening of the QE energies, while the latter are QE-plasmon ET rates, given by Eq. (8). Note that at resonance, we have δω⁽ⁱ⁾ = 0, while are given by Eq. (9).

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 the complex eigenvalues λ characterize the energy shifts and decay rates of the collective states. It is now easy to see that the vector is an eigenstate of the matrix Eq. (10), and the corresponding eigenvalue has the form , where is frequency shift of the collective state and is its decay rate. At resonance ω = ω_η, we again have δω_Ν = 0, while the decay rate of the collective state ψ_Ν takes the form [compare to Eq. (9)]

where is the projected plasmon mode volume (7) at the ith QE position. Thus, the collective state described by ψ_Ν exchanges its energy with the plasmon mode cooperatively, i.e., at a rate equal to the sum of individual QE-plasmon ET rates.^41,53 As we show below, the CET rate Eq. (11) determines onset of the transition to strong coupling regime for QE ensembles coupled to plasmonic resonators.

Since the imaginary part of the eigenvalue λ_Ν 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 EXCHANGE

Consider first a single QE situated near a plasmonic resonator with its frequency ω₀ close to the plasmon frequency ω_m. 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 and . To simplify the analysis, we assume for now that QE and plasmon are in resonance, ω₀ = ω_m. Typically, in plasmonic systems, the intrinsic QE decay rate γ₀ is much smaller than either γ_m or γ_et, and so we disregard it here as well; however small, γ₀ does play a crucial role in the interference effects, such as Fano resonances,³⁹ but such phenomena are beyond our work’s scope. In the weak coupling regime, both polaritonic bands are centered at the same frequency ω_m but are characterized by different spectral widths,

where we used the relation (12). For , the expansion of Eq. (13) over the small parameter yields and , which represent, respectively, the decay rates of weakly-mixed plasmon and QE states.

In the strong coupling regime, both polariton bands are characterized by constant spectral width γ_m/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 γ_m/2, which corresponds to the condition .

The onset condition (15) implies that, for plasmonic resonators with high quality factor , the QE-plasmon energy exchange takes place on a much longer time scale than the optical period. Qualitatively, the energy exchange dynamics can be described by the damped oscillator equation, , where x characterizes the QE-plasmon energy balance while frequency is the two-way QE-plasmon ET rate. After a pulsed excitation, the energy balance undergoes damped oscillations with energy-exchange frequency

and damping rate γ_m/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 , which coincides with Eq. (15).

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 , given by (11). Accordingly, the ensemble QE-plasmon coupling g_N is related to the CET rate as [compare to Eq. (12)]

where is the average plasmon mode volume for QE ensemble, defined as

The ensemble QE-plasmon coupling (17) is related as to the cooperative Purcell factor characterizing enhancement of the CET-based cooperative emission rate.^53,56 For a QE ensemble, the transition onset to strong coupling regime is obtained with replacing the ET rate in Eq. (15) by the CET rate: . In the strong coupling regime, the energy balance between the collective state and resonant plasmon mode oscillates with energy-exchange frequency , which vanishes at the transition point, and the same damping rate γ_m/2 [compare to Eq. (16)].

The ensemble QE-plasmon coupling g_N is related to individual QE couplings , and, therefore, it depends sensitively on the plasmon LDOS variations over the region QEs are distributed in. For example, if the plasmon LDOS (and, accordingly, the mode volume) is nearly constant in the QE region, e.g., within dielectric core enclosed by a metallic shell, the individual couplings gi are approximately equal and, hence, the ensemble QE-plasmon coupling exhibits a cavity-like scaling, . However, in open plasmonic systems with large number of QEs distributed outside the plasmonic structure (see Fig. 1), the ensemble QE-plasmon coupling saturates to a universal value, as we show later in this paper.

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 TIP

The 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 , so the coupling (12) has the form

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 l_m relative to the skin penetration length l_s (about 20 nm for Au in plasmon frequency domain). For small structures with l_m < l_s, the QE-plasmon coupling scales as , while in opposite case l_m < l_s, the metal volume V_met should be replaced by the effective volume the plasmon field is largely confined to. In either case, there is a significant enhancement of g relative to the QE coupling to a cavity mode, , so in the analysis below we use the notation V_met throughout. Importantly, in addition to this geometric volume effect, there is also strong field enhancement due to “lightning rod” effect near sharp metal tips or surface irregularities.

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 , we obtain

where is the plasmon field outside the structure normalized by its value at the tip. For good Drude metals in the plasmonic frequency range, we have , and the coupling (21) takes the form

The maximal value of g_tip is achieved in a close proximity to the tip, where . Comparing this value to the coupling g_cav for a QE inside a cavity, given by Eq. (1) with , we obtain the enhancement factor,

indicating that the plasmonic enhancement is due to both the geometric volume effect and the plasmon field enhancement characterized by . Note that for metal nanostructures with characteristic size larger than the skin penetration length 1_s, the metal volume in Eq. (23) should be replaced by , as discussed above.

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 , normalized by , plotted against normalized distance d/a for different values of aspect ratio a/b at fixed nanorod length a, i.e., for different metal volumes V_met = ab². It is clearly seen that for a/b > 1, i.e., when the system possesses hot spots near nanorod tips, the QE-plasmon coupling scales as with its overall magnitude falling off rapidly away from the tip. Note that while the coupling is largest at the tip , it is expected to saturate below QE-tip distances as the nonlocal effects become important.^57–59 In noble metals, this length scale is < 1 nm in the plasmonic frequency range and, therefore, the distance dependence in Fig. 3 is cut off below d/a = 0.05.

5.

SATURATION OF EXCITON-PLASMON COUPLING IN OPEN SYSTEMS

In 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 in Eq. (24) is independent of V₀. In this case, the QEs saturate the entire plasmon mode volume, i.e., beyond the QE region, the plasmon LDOS is negligibly small, so that, with expanding QE region, the ensemble QE-plasmon coupling (24) saturates to some value g_s. Note that, since the coupling (24) is independent of losses, such saturation is due to reduction of QE-plasmon ET rates away from the plasmonic structure, rather than any dephasing processes.

Let us now show that the actual value of saturated coupling g_s 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 , and we arrive at a universal saturated QE-plasmon coupling,

Using the relation for good Drude metals, the coupling (25) simplifies to

and the transition onset to strong coupling regime for large ensembles, , can be presented as

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 g_s 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 V_cav given by⁹

where is the cavity dielectric function. After averaging Eq. (28) over the cavity volume V_cav, we obtain , and the saturated coupling parameter for microcavities takes the form

Comparing to Eq. (26), we conclude that, for same QE concentrations, the saturated coupling for open plasmonic system is of the same order as for microcavities, implying that large Rabi splitting observed in open plasmonic systems are due to high QE concentrations. Note also that, due to much higher microcavity quality factors, the transition onset is lower as well.

6.

CONCLUSIONS

In 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. By defining accurately the plasmon mode volume, we have shown that for a QE near a sharp metal tip, the QE-plasmon coupling is enhanced by the factor relative to the QE coupling to a cavity mode, where V_met is metal volume that confines the plasmon field. For an ensemble of N QEs placed in a region with weakly varying plasmon LDOS, the QE-plasmon coupling exhibits cavity-like scaling . However, in open plasmonic systems, the ensemble QE-plasmon coupling saturates to a universal value g_s that does not depend on plasmonic system’s size and shape except indirectly via the plasmon frequency. Finally, we compared coupling parameters for QEs coupled plasmon and cavity modes to find that, for same QE concentrations, the open plasmonic systems offer no significant enhancement, implying that the observed large Rabi splittings are likely due to high QE concentrations in such systems.