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Deep strong light–matter coupling in plasmonic nanoparticle crystals


In the regime of deep strong light–matter coupling, the coupling strength exceeds the transition energies of the material1,2,3, fundamentally changing its properties4,5; for example, the ground state of the system contains virtual photons and the internal electromagnetic field gets redistributed by photon self-interaction1,6. So far, no electronic excitation of a material has shown such strong coupling to free-space photons. Here we show that three-dimensional crystals of plasmonic nanoparticles can realize deep strong coupling under ambient conditions, if the particles are ten times larger than the interparticle gaps. The experimental Rabi frequencies (1.9 to 3.3 electronvolts) of face-centred cubic crystals of gold nanoparticles with diameters between 25 and 60 nanometres exceed their plasmon energy by up to 180 per cent. We show that the continuum of photons and plasmons hybridizes into polaritons that violate the rotating-wave approximation. The coupling leads to a breakdown of the Purcell effect—the increase of radiative damping through light–matter coupling—and increases the radiative polariton lifetime. The results indicate that metallic and semiconducting nanoparticles can be used as building blocks for an entire class of materials with extreme light–matter interaction, which will find application in nonlinear optics, the search for cooperative effects and ground states, polariton chemistry and quantum technology4,5.

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Fig. 1: Plasmon polaritons in metal nanoparticle fcc crystals.
Fig. 2: Synthesis of gold nanoparticle crystals.
Fig. 3: Deep strong light–matter coupling in nanoparticle crystals.
Fig. 4: Counter-rotating and photon self-interaction terms in deep strong coupling.

Data availability

The measured and simulated optical spectra that support the current study are available at Source data are provided with this paper.


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We are grateful to A. Knorr, G. Weick and S. De Liberato for discussions and comments. We thank M. Wolf for the use of the supercontinuum light source. This work was supported by the European Research Council (ERC) under grant DarkSERS (772108). S.R. acknowledges the Focus Area NanoScale of Freie Universität Berlin. F.S. was supported by the DFG via project SCHU 3019/2-1. H.L. and F.S. acknowledge support by the Cluster of Excellence ‘Advanced Imaging of Matter’ of the DFG (EXC 2056, project 390715994). B.G.M.V. acknowledges the Coordenação de Aperfeiçoamento de Pessoal de Nivel Superior (CAPES) for financial support under the programme PDSE (grant number 88881.134611/2016-01) and Dahlem Research School (DRS). E.B.B. and B.G.M.V. acknowledge financial support from CNPq, CAPES (finance code 001) and FUNCAP (PRONEX PR2-0101-00006.01.00/15).

Author information

Authors and Affiliations



S.R. and N.S.M. conceived and designed the research project. The gold nanoparticle crystals were synthesized by F.S. and H.L.; the characterization and optical experiments were performed by N.S.M., Y.O., F.S. and S.J. The analytical theory and FDTD simulations were developed by N.S.M., B.G.M.V., E.B.B. and S.R. The data were analysed by N.S.M., F.S. and S.R.; S.R. and N.S.M. wrote the initial draft of the manuscript, on which all co-authors commented.

Corresponding author

Correspondence to Stephanie Reich.

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The authors declare no competing interests.

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Peer review information Nature thanks Junichiro Kono, Salvatore Savasta and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Plasmon dispersion without light–matter interaction, calculated with a microscopic theory based on the point-dipole approximation and the Drude model24 (see Methods).

The following parameters were used: fcc crystal, nanoparticle diameter d = 30 nm, interparticle gap size a = 2 nm, ħωp = 9.07 eV, ε = 1 and εm = 1.96. The plasmonic band structure was calculated using \({\omega }_{{\rm{pl}}}({\bf{k}})={\omega }_{{\rm{NP}}}\sqrt{1+{\rm{g}}f({\bf{k}})}\), where we set ft = −4/3 and fl = 8/3 close to the Γ point because of a divergence of the lattice sums; see discussion in ref. 24. The oscillations close to Γ are an artefact of the calculations and were omitted when calculating the polariton dispersion in Fig. 1b.

Source data

Extended Data Fig. 2 SEM images of gold nanoparticle crystals with large crystallites.

a, Gold nanoparticle crystal consisting of 30-nm gold nanoparticles. b, Gold nanoparticle crystal consisting of 59-nm gold nanoparticles. The scale bars are 1 μm.

Extended Data Fig. 3 Simulated optical transmission of aluminium nanoparticle fcc crystals.

The crystals consist of six layers of d = 30 nm nanoparticles with gaps a = 2 nm (black), 5 nm (cyan) and 10 nm (magenta). The dielectric constant of the embedding medium was set to εm = 1.96 and the optical properties of aluminium were modelled using a fit of the experimental data from Rakić49.

Source data

Extended Data Fig. 4 Approximations of the Hopfield Hamiltonian.

a, Vacuum Bloch–Siegert shift, ΔVBS, calculated from the difference between the polariton dispersion of the full Hamiltonian (black line in Fig. 4a) and the dispersion of the Hamiltonian when neglecting all counter-rotating terms (orange line in Fig. 4a). b, Fitting of experimental data (d = 59 nm) with the polariton dispersions of the interaction Hamiltonians discussed in Supplementary Information. The black line is a fit with the Hopfield model (Supplementary Information equation (S8)). The pink line shows a fit with the polariton dispersion when neglecting the A2 term (Supplementary Information equation (S9)). The orange line is a fit with the dispersion when neglecting all counter-rotating (CR) terms (no CR terms; Supplementary Information equation (S11)). The blue line is a fit with the RWA approximation (no CR terms and no A2 term; Supplementary Information equation (S12)).

Source data

Extended Data Fig. 5 Mean population of the ground state with virtual photons and virtual plasmons as a function of the wave vector k.

The populations were calculated from the Hopfield model following ref. 1 for the experimental nanoparticle crystals in Fig. 3. We used the fitting parameters shown in Table 1. Shown are results for crystals with d = 25 nm (orange), 30 nm (black), 41 nm (blue), 46 nm (yellow) and 59 nm (pink).

Source data

Extended Data Fig. 6 Absorption spectra calculated from the Hopfield Hamiltonian for ultrastrong to deep strong coupling.

The absorption by the lower polariton branch increases in intensity and decreases in line width, and vice versa for the upper polariton branch. The normalized coupling strength is η = 0.52.0 and varies in increments of ∆η = 0.05 (ħωNP = 4.06 eV, \({\omega }_{{\rm{pl}}}^{{\rm{t}}}=\sqrt{{\omega }_{{\rm{NP}}}^{2}-4{\varOmega }_{{\rm{R}}}^{2}/3}\), slab thickness 100 nm and γ = 5 × 1013 rad s−1). The model is discussed in Methods.

Source data

Extended Data Fig. 7 Evaluation of nanoparticle diameter d and interparticle gap a using a TEM image of a 59-nm nanoparticle monolayer.

a, TEM image of a monolayer of gold nanoparticles. b, Evaluation of nanoparticle sizes and interparticle gap sizes of the image in a using the software Image J. The software determines the size of the nanoparticles (coloured areas), giving the mean nanoparticle diameter. The interparticle gaps sizes were obtained by subtracting the particle diameters from the nearest-neighbour distances (white lines). c, Histogram of the nanoparticle diameters and fit with a Gaussian distribution. The mean values from the analysis of the TEM image in a are d = 60.6 nm and a = 1.5 nm. The scale bars are 100 nm.

Extended Data Fig. 8 Simulated plasmon-polariton dispersions for gold nanoparticle fcc crystals with different diameters d and interparticle gap sizes a.

The labels show d and a in nanometres. a, Dispersions for different nanoparticle diameters and an interparticle gap size of 2 nm. b, Dispersions for different gap sizes and a nanoparticle diameter of 50 nm. The data points were obtained from simulated absorption spectra and the solid lines are fits with equation (1) using ΩR and \({\omega }_{{\rm{pl}}}^{{\rm{t}}}\) as fitting parameters.

Source data

Extended Data Fig. 9 Comparison of polariton dispersions from the microscopic model (solid black lines) and FDTD simulations (data points).

a, The polariton dispersion shown in Fig. 1b with d = 30 nm, a = 2 nm and εm = 1.96. b, Parameters as in a, except for εm = 1. c, Polariton dispersion for d = 30 nm, a = 20 nm and εm = 1.96. We used the Drude model as the dielectric function of the metallic nanoparticles with ε = 1, ħωp = 9.07 eV and ħγp = 40 meV.

Source data

Extended Data Fig. 10 Effect of crystal thickness on the coupling strength.

a, Simulated polariton dispersion for an fcc gold nanoparticle crystal with d = 50 nm, a = 2 nm and εm = 1.96. The data points were obtained by fitting the absorption spectra of crystals with different numbers of layers. The black line is a fit of the data points for 20 layers with the Hopfield dispersion (equation (1)). The inset shows the normalized coupling strength as a function of layer number. b, As in a, but for the experimental data of the gold nanoparticle crystal with d = 41 nm. The black line is a fit of the data points for 5–7 layers.

Source data

Supplementary information

Supplementary Information

The Supplementary Information contains a Supplementary Table with all variables used in the manuscript and Supplementary Discussion on the Hopfield model.

Source data

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Mueller, N.S., Okamura, Y., Vieira, B.G.M. et al. Deep strong light–matter coupling in plasmonic nanoparticle crystals. Nature 583, 780–784 (2020).

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