## Introduction

Light-matter interaction between an optical mode and a quantum emitter is accurately described in terms of the cavity quantum electrodynamics (cQED) formalism1. The interaction can be weak or strong, depending on circumstances, leading to drastically different behavior. The regime of strong light-matter coupling, unlike its weak counterpart, leads to formation of hybrid cavity-emitter eigenmodes manifested in coherent energy exchange between the subsystems occurring on timescales that are much faster than the corresponding damping rates. Thus, the emitter and the cavity form a unified light-matter hybrid polariton, whose properties, including spontaneous emission and chemical potential, can be tuned2,3,4. Because of their compositional nature, polaritons are useful for photon–photon interactions, which lead to remarkable nonlinear and quantum optical phenomena, including photon blockade and Bose-Einstein condensation of exciton polaritons5,6,7. On the other hand, strong light-matter coupling may lead to changes in emitter properties, including photochemical rates8,9,10,11,12 and exciton transport13,14.

Traditionally cavity-exciton polaritons have been realized in atomic or solid state systems utilizing high quality-factor optical cavities at low temperatures15. Polaritonic behavior associated with “dressing” of the emitter by a cavity field is usually captured by traditional quantum optical approaches such as Jaynes–Cummings or Dicke models2,3,4,16,17. However, these quantum optical formalisms treat matter in an extremely simplified manner, that is, as a two-level system, leading to oversimplifications and inconsistencies in the description of the material subpart.

Advanced theoretical techniques developed recently allow for more sophisticated effects including, for example, multiple electronic resonances, accounting for atomic vibrations, and light-matter interactions beyond the point-dipole approximation. Significant progress along these lines has been achieved by several groups using various quantum optical and quantum chemistry methods9,10,11,18,19,20,21. However, typically either molecules or electromagnetic fields in these approaches are treated in a simplified manner.

Plasmon-molecule interactions can be modeled computationally using density-functional theory (DFT)22,23 and/or time-dependent density-functional theory (TDDFT)24 approaches, thanks to the relatively small number of atoms involved in these interactions. Here, we model the entire plasmon-exciton system by TDDFT, which allows the whole system to be treated on the same footing, enabling one to track effects related to modification of the matter subpart, which are inaccessible by purely quantum optical or classical electromagnetism methods. In fact, analytical and numerical quantum chemistry approaches, including DFT and TDDFT, have been successfully applied to study plasmon-molecule interactions, charge transfer, chemical enhancement and electromagnetic effects in surface-enhanced Raman scattering experiments25,26,27,28,29,30,31,32,33. However, vacuum Rabi splitting and strong plasmon-molecule coupling have not been in the focus. Yet, many recent experimental observations cannot be easily explained with current theories34,35,36.

In response to the above, quantum electrodynamics density-functional theory (QED-DFT) has been developed recently19, enabling the modeling of strong coupling between electromagnetic cavity modes and electronic excitations37,38. While the commonly-used TDDFT approaches with classical description of electric fields may not be able to describe the full range of quantum optical phenomena that QED-DFT aims at, we demonstrate in this paper that the cavity mode created by a localized surface plasmon resonance as well as its strong coupling with excitons can be described already within the standard TDDFT. In the following, we argue that theoretical predictions obtained at the nanometer scale may in fact be useful for understanding and modeling of more extended systems. We use TDDFT to study the optical response of aluminum (Al) nanoparticles composed of a few hundred atoms coupled with one to eight benzene molecules. Thereby we show that cavity quantum electrodynamics holds down to resonators on the order of a few cubic nanometers. In the case of an Al201 particle we observe a single-molecule coupling strength of 200 meV due to a very large vacuum field of 4.5 V · nm−1, while for a large number of molecules, the molecules perturb the system increasing the mode volume and decreasing the per-molecule coupling strength. For larger Al586 and Al1289 nanoparticles the coupling strengths are smaller and follow well the $$\sqrt N$$ dependence of coherent coupling with the number N of molecules.

## Results

### Modeling strong coupling with TDDFT

We model light-matter interactions by employing the real-time-propagation (RT) TDDFT approach39 based on the localized basis sets40,41 as implemented in the open-source GPAW package42,43. This RT-TDDFT code is combined with extensive analysis tools44 that are utilized for analyzing the electron-hole transition contributions to resonances and visualizing them as transition contribution maps (TCM)44,45. See Methods for detailed description. We consider a proof-of-principle plasmonic system based on idealized Al nanoparticles. The free-electron-like electronic structure of Al greatly simplifies the analysis in contrast to noble metals with d-electron-screened plasmons. In addition, Al has attracted recent interest as an alternative plasmonic material46.

Our model systems consist of Al nanoparticles (regular truncated octahedra with 201, 586, and 1289 atoms, approximate diameters of 1.8, 2.7, and 3.6 nm, respectively) and benzene molecules (see Supplementary Fig. 1). The Al201 nanoparticle exhibits a plasmon resonance at ωAl = 7.7 eV (Fig. 1a), the collective nature of which is recognizable in the TCM (Fig. 1b)44. The benzene molecule has a doubly-degenerate resonance at ωB = 7.1 eV with a transition dipole moment μ1 = 4.35 D (Fig. 1a and Supplementary Fig. 2).

When the nanoparticle and benzene molecules are placed in proximity (bottom part of Fig. 1a) their resonances couple and form two polaritons. The lower (at 6.9 eV) and upper (at 7.7 eV) polaritons are, respectively, mixed symmetric and antisymmetric plasmon-molecule states as apparent in the TCMs of the coupled system (Fig. 1d, e). A comparison with the TCMs of the uncoupled constituents (Fig. 1b, c) reveals the individual contributions to the polaritons. The low-energy (2 eV) transitions near the Fermi level form the plasmon, while the contribution from about −2 to 3 eV originates from the molecular exciton. Crucially, the two polaritonic states in Fig. 1d, e differ with respect to the sign of the molecular contributions. In case of the lower polariton (LP) plasmonic and molecular transitions are in-phase, while for the upper polariton (UP) they are out-of-phase. This, respectively, symmetric and antisymmetric combination is also clearly visible in the induced densities (Fig. 1a). At the LP the dipoles of both particle and molecules are parallel, while they are antiparallel at the UP. Such an arrangement is archetypal for a strongly coupled system. This observation is valid also for the other systems discussed below, indicating the presence of strong or near-strong coupling already in the case of a single molecule placed 3 Å away from the Al201 particle. We note here, that the density of states of the molecular elements in the coupled system does not significantly differ from its uncoupled counterpart (Supplementary Fig. 3).

To determine the fidelity of TDDFT for modeling strong coupling, we calculated the photoabsorption spectra of Al201, Al586, and Al1289 coupled to N benzene molecules, whose dipole moment is aligned collinearly with that of the plasmon vacuum field. The molecules are placed at the corners of two opposing {100} facets of the particles (Supplementary Fig. 1) and structural relaxation is neglected for simplicity. The coupled systems exhibit the LP and UP modes, the splitting of which increases with the number of molecules, see Fig. 2 (TCMs show similar symmetric/antisymmetric mixtures of plasmonic and molecular transitions, see Supplementary Fig. 4). The Rabi splitting of the absorption spectra for Al201 and one benzene molecule equals 730 meV, a value comparable to the plasmon width. A non-negligible part of the UP/LP separation originates, however, from the plasmon-exciton detuning and the coupling strength g is 200 meV for one benzene, a value smaller than the geometrical mean of the plasmon and exciton widths (300 meV). The strong coupling condition is, however, fulfilled for N ≥ 3 for Al201 and for larger N for larger particles.

### Coherent coupling and mode volumes

One signature of collective coherent coupling is the square-root dependence of g on the number N of identical excitons with transition dipole moment μ1 interacting with an optical cavity in the standard QED expression,

$$g(N) = \sqrt N {\rm{\mu}} _1E_{{\mathrm{vac}}},$$
(1)

where $$E_{{\mathrm{vac}}} = \sqrt {\hbar \omega _{{\mathrm{Al}}}/(2\varepsilon _0V)}$$ is the vacuum field and V is the volume of the electromagnetic mode15. Using the Al201 particle as an example, the expected scaling is shown in Fig. 3 by the black dotted line using μ1 of a benzene molecule and $$E_{{\mathrm{vac}}}^{(201)} = 4.5\,{\mathrm{V}}\, \cdot \,{\mathrm{nm}}^{ - 1}$$ with ωAl = 7.7 eV and V = 3.3 nm3 corresponding to the volume of the Al201 particle, which is an adequate estimate for a nanometer-sized particle47. To estimate the coupling strengths g from our first-principles data, we fit the absorption spectra with a coupled harmonic oscillator model48 (see Supplementary Note 1), yielding the g values shown as symbols in Fig. 3. We notice that the coupling strength values obtained using the standard QED expression of Eq. (1), which is insensitive to spatial variations in the electric field, are larger than the ones derived from coupled oscillator fitting. A similar observation holds for the larger particles Al586 and Al1289, except, that due to larger volumes, their respective vacuum fields $$(E_{{\mathrm{vac}}}^{(586)} = 2.6\,{\mathrm{V}} \cdot {\mathrm{nm}}^{ - 1},\,E_{{\mathrm{vac}}}^{(1289)} = 1.8\,{\mathrm{V}} \cdot {\mathrm{nm}}^{ - 1})$$ and coupling strengths are reduced.

### Coupling efficiencies

Equation (1) is accurate under the condition that all excitonic modes are coherently coupled with the same, maximal rate to the cavity. This is true for structures such as Fabry-Pérot cavities, photonic crystal slabs, or micropillars, in which the anti-node of the mode is accessible to excitonic modes of molecules or semiconductors by precise placement via trapping or doping3,15. For isolated plasmonic particles the maximum of its mode is, however, typically inside the particle47, such as in the present case. Consequently, the molecular coherent dipole moment interacting with the cavity is smaller than its maximum value of 4.35 D, and it is estimated to be in the range from 2 to 4 D, increasing with the size of the nanoparticle. The reduction of the coherent dipole moment is typically expressed in terms of an efficiency factor η determined by the ratio of the mode energy density at the position of the exciton to its maximum value49.

Before extracting efficiency factors η from the TDDFT data, we estimate their values on the basis of classical electromagnetic calculations, which are acceptable at a semi-quantitative level for picocavities50, using a local Drude-permittivity tailored to match the TDDFT absorption spectrum for Al201. We find that in the spatial region occupied by the benzene molecule the efficiency η ranges from 0.3 to 0.5 (Fig. 4a), due to the rapid decrease of the plasmon-induced electric field. However, such a low value of η predicted by calculating the mode profile of a bare cavity might not hold in practice. It is known that the presence of a material with a refractive index greater than the background causes additional field localization, increasing the coupling strength49. While a single benzene molecule cannot be meaningfully described in terms of a refractive index, it causes a similar effect. This is supported by the electric near-field enhancements calculated with TDDFT (Fig. 4b–d). In comparison to the field of the plasmon in the bare nanoparticle, in the LP/UP of the coupled system the benzene molecule focuses the electric field, modifying the cavity and, consequently, its vacuum field49.

To estimate the coupling efficiencies η from the TDDFT-derived coupling strengths (Fig. 3), we extract their values as modifications of the slope predicted by Eq. (1). The resulting dependencies $$g = \eta \sqrt N {\rm{\mu}} _1E_{{\mathrm{vac}}}$$ are shown as dashed lines in Fig. 3. For Al201, the corresponding value η201 = 0.47 gives a good agreement for N ≤ 3, but for N > 3 the coupling strengths fall below the slope set by η201. Such a deviation for large N is not observed for the two larger particles, which, respectively, have efficiency factors of η586 = 0.61 and η1289 = 0.78. These larger efficiencies imply that the plasmonic modes of the larger particles are used more efficiently than in the case of Al201. Minor deviations seen for the two larger nanoparticles for N < 3 are probably caused by inaccurate fitting of the absorption spectra as the LP and UP are not clearly separated.

To test the effect of a diminishing overlap between the vacuum field of the cavity and the molecular exciton, we calculated the dependence of coupling strength on the nanoparticle-molecule separation (Fig. 5). The Rabi splitting of LP and UP decreases with increasing separation and the reduction of the coupling efficiency49 leads to a decrease in the coupling strength (Fig. 5b).

## Discussion

The coupling efficiencies estimated from classical electromagnetic and TDDFT calculations are consistent. The addition of molecules leads to a systematic red shift of the plasmon (Supplementary Fig. 5), which is largest for Al201 and scales approximately inversely with particle volume in agreement with earlier predictions51. The red shift happens simultaneously with strong coupling and complicates the analysis. While the calculations yield a $$\sqrt N$$ dependence of the coupling strength for the larger particles (Al586, Al1289), for the smallest particle (Al201) the efficiency varies with the number of benzene molecules for N 3. In the Al201 case, the volume occupied by the molecules is non-negligible relative to the particle volume and every additional molecule changes the properties of the nanoscale cavity and subsequently modifies the coupling.

From the perspective of classical electromagnetism, the molecules can be represented as a polarizable background with excitons modelled by Lorentzians52. This background can be treated as part of the cavity and consequently modifies its local optical density of states (LDoS)49,53. This modification manifests itself as focusing of the electric field around the molecules, clearly visible for Al201 coupled with one benzene molecule (Fig. 4c, d), in comparison to the resonance of Al201 alone (Fig. 4b). These LDoS modifications increase the mode volume, decreasing (or at least not enhancing) the coupling strength per molecule. Such changes of the cavity induced by adding molecules are not visible for larger particles, as for Al586 and Al1289 the coupling strengths follow the $$\sqrt N$$ dependence.

In conclusion, we demonstrated the suitability of first-principles TDDFT for studying strong coupling between plasmons and molecular excitons. This approach allows us to capture relevant interactions at the atomic level. This is important for studying coupling of molecules to picocavities slightly larger than the molecules themselves50,54 as well as ultrastrong coupling that leads to modification of the molecular ground state18,55. Furthermore, we have shown the degree to which the simple cQED description holds for small systems. For small single-particle cavities, such as the Al201 particle considered here, the presence of molecules modifies the cavity and, consequently, its mode volume. The mode volume, in turn, affects the coupling strength, whose value does not increase as quickly as predicted by QED. This slower increase of g is noticeable for the Al201 particle, but for larger sizes (Al586, Al1289) deviations from Eq. (1) become negligible. Despite the deviations between a simple cQED formula and TDDFT calculations, the order of magnitude agreement between them is quite remarkable. In particular, it is rather surprising that such a simple formula at all holds at such small scales. Based on the obtained coupling strengths, calculated mode volumes and field enhancements, reaching single/few-molecule ultrastrong coupling for single-particle cavities may prove challenging.

## Methods

### DFT and TDDFT calculations

The DFT and TDDFT calculations were carried out using the Perdew-Burke-Ernzerhof (PBE)56 exchange-correlation functional in the adiabatic limit. The spectra were calculated using the δ-kick technique39 in the linear-response regime and employing the dipole approximation for light-matter interaction. The default projector augmented-wave57 data sets and double-ζ polarized (dzp) basis sets provided in GPAW were used. The dzp basis set of Al includes diffuse 3p functions, which are important for describing plasmon resonances58. To minimize spurious effects due to the basis-set superposition error, the so-called ghost-atom approach was used separately for each nanoparticle size to keep the total system basis set as intact as possible despite the changing number of surrounding molecules. In general, while the used basis sets might not be adequate for yielding numerical values at the complete-basis-set limit, they are expected to be sufficient for the purposes of the present work. A grid spacing parameter of 0.3 Å was chosen to represent densities and potentials, and the molecules/particles were surrounded by a vacuum region of at least 6 Å. The Hartree potential was evaluated on a larger grid with at least 120 Å vacuum around the system and a coarser grid spacing of 1.2 Å, and subsequently refined to the original grid. For the time propagation, we used a time step of Δt = 15 as and total propagation time of at least T = 30 fs. The spectra were broadened using Lorentzian spectral broadening with η = 0.1 eV corresponding to a full width at half-maximum of 0.2 eV. Codes for reproducing the calculations are available (see Code availability).

### Transition contribution map (TCM)

A TCM is used for visualizing the Kohn-Sham (KS) electron-hole transition contributions to photoabsorption. Briefly, the photoabsorption cross-section S(ω) is expressed in the basis of occupied (i) and unoccupied (a) KS states as $$S(\omega ) = \mathop {\sum}\nolimits_{ia} S_{ia}(\omega )$$44. In TCM, the elements of the matrix Sia(ω) at a chosen resonance energy are plotted on a Gaussian-broadened two-dimensional plane spanned by the energy axes for occupied and unoccupied KS states. The color indicates the sign of the transition contribution: positive (red) or negative (blue), and the color intensity indicates the magnitude of the contribution. The maps are augmented with the corresponding densities of states, and a diagonal line is drawn to indicate the KS eigenvalue difference corresponding to ω. See ref. 44 for a detailed description of TCM construction.

### Software used

DFT calculations were carried out using the GPAW package42,43 with localized basis sets (LCAO mode)41. TDDFT calculations were conducted using the LCAO-RT-TDDFT implementation in GPAW40 and analyzed using the TCM tools therein44. The ASE library59 was used for constructing the atomic structures. The NumPy60 and Matplotlib61 Python packages, the VMD software62,63, and Inkscape were used for processing data and generating figures.