Collective multimode strong coupling in plasmonic nanocavities

2.1 Quantum oscillations in multimode strong coupling

The oscillations shown in Figure 1c arise from the off-resonant modes but are hard to interpret due to the high plasmonic losses. To overcome this, we initially assume the system evolves in the absence of loss (i.e. κ _ξ = 0) such that the evolution is unitary and governed by the Schrödinger Equation i ∂ t | ψ 〉 = H | ψ 〉 where the excited state population of the QE can be determined both numerically and analytically. This helps provide a comprehensive and in-depth understanding of the interactions involved. To derive an equation for the QEs excited state population, we first transform to the interaction picture such that i ∂ t | ψ ̃ 〉 = V int | ψ ̃ 〉 where V int = e i H 0 t H int e − i H 0 t and | ψ ̃ 〉 = e i H 0 t | ψ 〉 . In fact, due to conservation of the excitation number, we can also express the quantum state as | ψ ̃ 〉 = c 0 | 0 , e 〉 + ∑ j c ξ a ξ † | 0 , g 〉 with amplitudes c ₀ and c _ξ , respectively. From this, we obtain two coupled equations of motion:

where Δ _ξ = ω ₀ − ω _ξ is the detuning between the QE and mode ξ. To solve Eq. (3) and Eq. (4) for c ₀(t), we take their Laplace Transform, where we define c ̃ ( s ) = L c ( t ) and use the identities L ∂ t x = s x ̃ ( s ) − x 0 ( 0 ) and L f ( t ) e α t = f ̃ ( s − α ) . Hence, two new coupled equations are obtained given by s c ̃ 0 ( s ) − c 0 ( 0 ) = − i ∑ ξ g ξ c ̃ ξ ( s − i Δ ξ ) and s c ̃ ξ ( s ) − c ξ ( 0 ) = − i g ξ c ̃ 0 ( s + i Δ ξ ) , respectively, which we then solve algebraically to give:

where c ₀(0) = 1 is the initial population of the QEs excited state. The key step in solving Eq. (5) is to write the expression in square brackets as a quotient of two polynomial functions:

where

Importantly, when Δ _i ≠ Δ _j for all i ≠ j, then P⁽ⁿ⁺¹⁾(s) has n + 1 distinct and purely imaginary roots iλ _j where λ _n < λ _n−1 < … < λ ₁ < λ _n+1 (see Proof of distinct roots section in Supplementary information for more details). This enables us to factorise P⁽ⁿ⁺¹⁾(s) and rewrite Eq. (5) in a form that has a simple inverse Laplace Transform:

where α j = Q i λ j / d P ( n + 1 ) d s | i λ j . Finally, taking the inverse Laplace Transform of Eq. (9) now yields an expression for c ₀(t), from which we find the QEs excited state population:

where Ω _j = λ _j − λ _n+1 and, therefore, gives the oscillation frequencies present in the quantum dynamics of a multimode system in the limit of zero loss. In fact, Eq. (10) shows that there are up to n(n + 1)/2 frequency components in the excited state population of a QE strongly coupled with n modes. More specifically, there are n components corresponding to the mode frequencies |Ω _j | and n(n − 1)/2 corresponding to the interference terms |Ω _j − Ω _i |. Importantly, the oscillation frequencies |Ω _j − Ω _i | arise from non-interacting modes (of a Lorentzian form) and, therefore, are not equivalent to the direct coupling between modes as seen in hybrid metallodielectric cavities (that may exhibit modes of non-Lorentzian form).

The oscillation frequencies |Ω _j | and |Ω _j − Ω _i | – which are calculated through the roots of Eq. (7) – are shown in Figure 2 in red and green dots, respectively. They are found to show excellent agreement with the full numerical calculations, which show the Fourier Transform (FT) of the quantum dynamics calculated with no loss (κ _ξ = 0) through unitary evolution (black lines) and with loss (κ _ξ ≠ 0 obtained from the QNM calculations for each mode) through integration of Eq. (1) (purple dashed lines). Intermediate steps with l _max < 9 illustrate the effect of gradually including more modes coupled to the QE. Such a systematic study elucidates the origin of each oscillation frequency and, therefore, helps in understanding the physics of multimode cavities governed by Eq. (1) or Eq. (10). In addition, note that the broadening of the peaks due to loss also has the effect that if the oscillation frequencies are spectrally close, it becomes difficult to resolve them in the spectra as seen in Figure 2.

Figure 2:

Frequency components in multimode strong coupling. The Fourier Transform n ̃ ( ω ) of the QEs excited state population when including interacting modes up to ξ = (ℓ _max0). The solid black lines are numerical results without loss (i.e. κ _ξ = 0) and the dashed lines with plasmonic loss (i.e. κ _ξ ≠ 0). The red and green dots result from |Ω _j | and |Ω _i − Ω _j | calculated from the roots of Eq. (7). The sub-figures show the frequencies for when (a) ℓ _max = 1, (b) ℓ _max = 2, (c) ℓ _max = 3, (d) ℓ _max = 5, (e) ℓ _max = 7, (f) ℓ _max = 9, respectively. Note, the spectra with loss are cut to remove the peak at ω = 0, which results from the FT of a damped oscillations.

We initially consider one plasmonic mode (ℓ _max = 1) as is often assumed to be the case and obtain the Rabi oscillation frequency |Ω₁| as expected. Considering two plasmonic modes (ℓ _max = 2) leads to three distinct peaks, one corresponding to each mode, i.e. |Ω₁| and |Ω₂| and one interference peak |Ω₁ − Ω₂| of smaller amplitude. Here, all three peaks have a high intensity and contribute to the QEs evolution, but the off-resonant mode is the most significant. This is not always the case, and as we see later depends on the number and properties of the strongly coupled modes (i.e. their coupling strengths and detunings). As we increase the number of modes strongly coupled with the QE, we continue to observe up to n(n + 1)/2 oscillations in the quantum dynamics. For a large number of modes (i.e. ℓ _max ≥ 5), the quantum dynamics are completely dominated by a single ultra-fast frequency component |Ω _n | with only a small contribution from other components. It is extremely important to note, that although we associate the frequency component |Ω _j | with the mode j, each peak still depends on all the other strongly coupled modes in the cavity. In fact, the |Ω _j |, mode j association is lost the more strongly coupled modes are considered, as each mode gradually shifts from being a distinct peak to contributing to the single ‘supermode’ frequency component |Ω _n |.

Note that this ultra-fast oscillation |Ω _n | emerges from the combined effect of all modes and does not occur with the same amplitude and frequency if only some of the modes are considered (see Supplementary information Figure S2). Therefore, this is a collective phenomenon and necessitates a full description of the plasmonic environment. We also note that the supermode behaviour described here arises from the collective and coherent response to multiple (potentially distinct) modes and is different from the apparent single mode behaviour arising from damping or dissipation, which prevents multiple modes that are spectrally close from being resolved. In addition, this ultrafast oscillation frequency – due to the collective multimode strong coupling – is also robust to displacements of the QE as shown in the Supplementary information Figure S7 and S8. This is due to the large field enhancements of the m ≠ 0 modes away from the centre.

2.2 Critical transition to collective strong coupling

The amplitudes of the oscillations at frequencies |Ω _j | and |Ω _j − Ω _i | depend on the magnitude of the coupling strengths g _j compared to the detunings Δ _j . To determine the impact of the coupling strength on the oscillations reported above, we change the dipole moment μ (which is an overall scale factor of the coupling strengths) and identify three distinct regions. In Figure 3a and b, the FT of the quantum dynamics is shown for a system with and without loss, respectively, for many dipole moments. For small dipole moments such that g _i < |Δ _i − Δ _i+1| for each mode (region I), the dynamics are determined by a single resonant mode and leads to a single (Rabi) oscillation. This applies to the majority of molecules commonly utilised in plasmonics, such as Cy5 and methylene blue, indicating that for QEs with small μ single mode strong coupling is reached. Note, however, that when plasmonic losses are present in this system, damping prevents single mode strong coupling even for the same dipole moments. For intermediate dipole moments (i.e. quantum dots) such that |Δ _i − Δ _i+1|² < |g _i |² ≪|Δ _i+1||Δ _i − Δ _i+1| (region II), then multimode strong coupling occurs. In this region, the off-resonant modes produce multiple distinct frequency peaks |Ω _j |, while the interference components |Ω _i − Ω _j | have a small amplitude and are hard to observe (further discussion is included in the Supplementary information Figure S3 for μ = 30 D). Finally, for large dipole moments such that |g _i |² ≫|Δ _i+1||Δ _i − Δ _i+1| (region III), collective multimode strong coupling occurs. While this is a multimode phenomenon, the quantum dynamics are now governed by a single ultra-fast ‘supermode’ oscillation |Ω _n |. This ‘supermode’ oscillation results in energy exchange over an order of magnitude faster than single mode strong coupling alone. In region II, the supermode peak |Ω _n | can have the largest amplitude at some dipole moments, but in region III, it completely dominates the quantum dynamics. The quantum dynamics in Figures 1 and 2 for μ = 72 D are within region III and, therefore, show the collective interaction with multiple off-resonant modes.

Figure 3:

Transition from single to collective multimode strong coupling. Fourier Transform of the QEs excited state population as a function of dipole moment μ. The QE interacts with modes up to ξ = (90) with (a) assuming no loss, i.e. κ _ξ = 0 and (b) including loss, i.e. κ _ξ ≠ 0. Three regions, dictated by the dipole moment, govern the QEs evolution (I) single mode strong coupling (II) multimode strong coupling and (III) collective multimode strong coupling. The single mode approximation is shown in the dashed red dashed line. The critical dipole moment is also shown and represents the onset of the ultra-fast collective strong coupling. The FT spectra in (b) are cut to remove the peak at ω = 0, which results from the FT of damped oscillations. Below 20 D, the |Ω₁| peak significantly overlaps with the ω = 0 peak and so cannot be removed successfully.

Crucially, a transition occurs at a critical dipole moment (μ _c), where the dominant frequency component switches from |Ω₁| to |Ω _n | as shown in Figure 3 (bottom panels). At the critical dipole moment, the maximum oscillation frequency in a lossless system increases from 34 to 294 THz at μ _c ∼ 27 D and from 75 to 360 THz at μ _c ∼ 38 D in an equivalent system with loss. The critical dipole moment occurs at higher values with loss because dissipation suppresses the amplitude of the off-resonant peaks and so requires larger coupling to reach criticality. In the limit of large coupling, the oscillation amplitudes α _i switch from behaving like 1 + λ i ∑ j = 1 n 1 / ( λ i − Δ j ) − 1 to 1 + λ i ∑ j = 1 n 1 / ( λ i − Δ j ) − ∑ j ≠ k g k 2 / ( ( λ i − Δ j ) ( λ i − Δ k ) ) − 1 (see Dependence of amplitudes on dipole moment section in the Supplementary information for more details). The former has the most dominant oscillations for λ ₁, i.e. the component oscillating at |Ω₁|, while the latter has the most dominant oscillations for λ _n , i.e. the component oscillating at |Ω _n |. This occurs because the interactions with each mode interfere constructively for the amplitude α _n when μ > μ _c , whilst there is always some destructive interference for the other amplitudes. Therefore, past the critical dipole moment (which occurs at some point between these two regions) energy exchange with the plasmonic nanocavity becomes ultra-fast, and represents a collective multimode strong coupling regime. This can also be observed in the mode populations, where: (1) below the critical dipole moment most modes oscillate out of phase with one another and (2) above the critical dipole moment they begin to oscillate in phase at the supermode frequency |Ω _n |. For more information and discussion, see Mode amplitudes section in the Supplementary information.

This transition can be observed experimentally by measuring the scattering cross section spectra of the system – which has resonances at the eigenenergies of the Hamiltonian. In a bare cavity (without the QE), these experimental spectral resonances occur at the mode frequencies ω _ξ . However, with the presence of a QE strongly coupled with just the resonant plasmonic mode (in region I), the (10) mode peak splits into two polariton branches with splitting |Ω₁| characteristic of single-mode strong coupling as previously reported experimentally [5]. For collective multimode strong coupling (region III), the largest splitting occurs between the lowest and highest eigenfrequencies, with splitting |Ω _n |. In the Supplementary information Figure S10, we present the eigenenergies of the Hamiltonian as a function of dipole moment, as a useful guide to experiments on the expected mode splitting, but one can obtain theoretically the full scattering cross-sectional spectrum via quantum simulations similar to those presented in [39] for a single mode. Nevertheless, we expect experimental spectra to have consistent linewidths, amplitudes and resonant frequencies as the results presented in Figures 2 and 3b.

In plasmonic systems, multiple plasmonic modes cannot be neglected; the single-mode approximation holds only for small dipole moments, where strong coupling is often unachievable in practical setups due to the large plasmonic losses. As the coupling strength increases, the interaction transitions into a multimode interaction and eventually to a fully collective regime characterised by a dominant ultra-fast oscillation governing the quantum dynamics. This behaviour highlights a key advantage of quantum dots (QDs) to reach the collective multimode strong coupling regime in quantum plasmonics; having dipole moments varying significantly and even surpassing 70D (as seen in In GaN/GaN QDs). As a result, QDs can produce ultra-fast oscillations nearly two orders of magnitude faster than those achievable with dye molecules, and one order of magnitude faster than single mode approximations. In addition, collective multimode strong coupling can also be attained in plasmonic systems of different shapes and with further reduced mode volumes, enabling critical coupling even at lower dipole moments. It should be noted that our results are general for any nanophotonic system that supports multiple Lorentzian modes, since their spectral proximity allows for multiple modes to be coupled to a QE. These findings present a novel pathway for ultrafast coupling in multimode systems, with significant potential for applications in quantum information and sensing.