Ultrastrong coupling between nanoparticle plasmons and cavity photons at ambient conditions

The study addresses achieving ultrastrong coupling (USC) between light and matter—defined when the coupling strength g exceeds about 10% of the transition energy ω—at ambient conditions. In USC, common approximations like the rotating wave approximation fail and terms such as fast-rotating and the quadratic A^2 must be included, leading to phenomena like coupling-dependent vacuum energy and virtual photon population in the ground state. Prior experimental USC realizations typically require cryogenic temperatures and strong magnetic fields (e.g., Landau polaritons, superconducting circuits) or fill nearly 100% of cavity volume (e.g., organic molecules, intersubband transitions). The paper proposes and investigates a platform based on an array of plasmonic gold nanorods placed at the antinode of an optical Fabry-Pérot microcavity to achieve room-temperature USC without magnetic fields. The objectives are to demonstrate large normalized coupling strengths (g/ω up to ~0.55) using only a small fraction of cavity mode volume (~4%), to analyze the system using the full Hopfield Hamiltonian, and to identify signatures of USC such as ground-state (vacuum) energy modification and finite photonic occupancy in the ground state.

USC has been realized in systems such as Landau polaritons and superconducting circuits (g/ω ≥ 1, deep strong coupling), but under cryogenic and high-field conditions. Room-temperature USC has been reported using organic molecules in microcavities with g/ω ~0.3, and intersubband transitions in doped quantum wells achieving g/ω ~0.7, typically by saturating the cavity volume with active material. Plasmonic lattices and single nanorods have shown strong coupling with cavity modes but had not reached the USC regime. Theoretical frameworks emphasize the need for full Hamiltonians including fast-rotating and A^2 terms to avoid unphysical predictions and to capture vacuum energy dependence on coupling. Prior works also discuss phenomena such as dynamical Casimir effect and single-photon frequency conversion arising from USC.

• System design: A sub-diffractive periodic array of gold nanorods is placed at the electric field antinode (center) of a Fabry-Pérot microcavity formed by two gold mirrors with a SiO2 spacer. The array couples to the cavity's fundamental FP mode to form plasmon–cavity polaritons. Even FP modes have a node at the center and interact weakly; odd modes strongly hybridize with the plasmonic array.
• Numerical simulations: Finite-difference time-domain (FDTD, Lumerical) simulations computed absorption spectra versus cavity thickness and field profiles. Simulations used a normally incident plane wave, polarization along the rods, periodic boundary conditions, gold permittivity from Palik, mesh Δr=4 nm with 2 nm refinement near metals. Results showed anticrossing between the 1st FP mode and plasmon resonance, strong blue-shifts of odd modes, and field distributions consistent with lower (antisymmetric) and upper (symmetric) polaritons. Estimated Rabi splitting near resonance (ωcav ≈ ωpl ~0.8 eV) reached ~1 eV.
• Sample fabrication: On glass coverslips, bottom Au mirror (10 nm) with 2 nm Cr adhesion layer was deposited by e-beam evaporation, followed by PECVD SiO2 half-cavities (various thicknesses). Gold nanorod arrays (lengths 200–400 nm, width 50 nm, height 20 nm, side-to-side spacing 30 nm; arrays 250 × 250 µm^2) were patterned by e-beam lithography atop half-cavities. The top SiO2 half and a 10 nm Au top mirror completed the cavity. Bare arrays on glass served as references. SEM used conductive polymer coating; morphology assessed with Zeiss SEM.
• Optical characterization: FTIR microscopy (Bruker Hyperion 2000 with Vertex 80v, NA 0.4) measured normal-incidence reflection and transmission over ~80×80 µm^2, 2 cm^-1 resolution, with polarization parallel/perpendicular to rods; reference mirror was planar Au. Absorption computed from R and T. Visible-range reflectivity for thin (100 nm) cavities used a 20× objective (NA 0.45) and a fiber-coupled spectrometer, normalized to a dielectric-coated Ag mirror.
• Hamiltonian modeling and fitting: The system was analyzed using the full Hopfield Hamiltonian (Coulomb gauge) including fast-rotating and A^2 terms: H = ħωcav(a†a+1/2)+ħωpl(b†b+1/2)+ħgc(a†+a)(b†+b)+ħgc(a†+a)^2. Fits used measured coupled-mode reflection dips versus bare cavity energy (from uncoupled cavity spectra) to extract plasmon frequency ωpl and coupling gc, focusing on the 1st FP mode and the collective long-axis plasmon mode. Single-mode and multimode Hopfield formulations were compared; both gave consistent coupling strengths when higher odd modes were detuned. A geometry-based estimate of gc used Evac = √(2ωc/ε0 Leff) and nanorod dipole moments inferred from scattering cross-sections via the Larmor formula.
• Ground-state analysis: Using extracted gc and ωpl, the ground-state energy shift ΔEG = (ω+ + ω− − ωcav − ωpl)/2 and photonic occupancy ρphot = ⟨G|a†a|G⟩ were calculated as functions of detuning.

• Ultrastrong coupling at room temperature: Experimental reflection/absorption spectra show clear anticrossing between the cavity's 1st FP mode and the nanorod plasmon with large splittings. At near resonance (e.g., ~500 nm cavity thickness), the measured vacuum Rabi splitting between reflection dips is ~0.8 eV at resonant energy ~0.7 eV, exceeding both bare cavity and plasmon frequencies.
• Coupling strengths from Hopfield fits: For Lrod = 300 nm arrays, fitting yields ωpl ≈ 640 meV and gc ≈ 300 meV, giving Rabi splitting Ωr = 2gc = 600 meV at zero detuning. Across nanorod lengths 200–400 nm, normalized coupling gc/ωpl ranged from 0.4 to 0.56, unambiguously in the USC regime; maximum reported g/ω ≈ 0.55.
• Small mode filling: The nanorod layer occupies only ~4% of the cavity volume, yet achieves USC, contrasting with platforms requiring near-100% filling.
• Mode hybridization: Odd FP modes strongly hybridize and are pushed to higher energies; even modes remain largely unperturbed due to an electric field node at the cavity center. The upper polariton crosses the even FP mode consistent with multimode Hopfield predictions and FDTD.
• Narrow polariton linewidths: Observed polariton bands are significantly narrower than the bare plasmon mode, qualitatively attributed to suppression of free-space radiative losses by the cavity; polariton linewidths are governed mainly by cavity leakage and non-radiative losses.
• Ground-state (vacuum) energy modification: Estimated ΔEG at zero detuning scales as ~g^2/(4ωpl); for g/ωpl ≈ 0.5, ΔEG ~75 meV (about ~12% of the uncoupled ground-state energy). Using measured polariton energies, normalized ground-state energy changes up to ~10% are inferred.
• Virtual photon population: The ground state acquires photonic occupancy up to ~0.06 photons near zero detuning (ωpl ≈ ωcav ≈ 0.5–0.7 eV), indicating a finite virtual photon component.
• Consistency with geometry-based estimates: Using estimated nanorod dipole moment (~3.4×10^4 Debye for 400 nm rods) and array density yields gc ≈ 0.3 eV, in excellent agreement with fits.
• Importance of full Hamiltonian: Neglecting the A^2 term and/or fast-rotating terms leads to unphysical spectra (imaginary or negative energies) and overestimated coupling, underscoring the necessity of the full Hopfield model.

The results directly answer the central question of whether ultrastrong light–matter coupling can be realized at ambient conditions with minimal cavity filling. By coupling a plasmonic nanorod array to a Fabry-Pérot microcavity, the study achieves gc/ωpl up to ~0.55 with only a single nanoparticle layer (~4% volume fraction). This demonstrates a practical route to high-oscillator-strength USC in the optical/near-IR range without cryogenics or magnetic fields. The observed vacuum energy modification (~10%) and finite ground-state photonic occupancy are hallmarks of USC beyond the rotating-wave approximation and naive coupled-oscillator models. The findings have implications for exploring USC-driven phenomena, such as dynamical Casimir radiation and vacuum-induced forces on mesoscopic objects. The platform affords tunability via nanorod length, array density, cavity thickness, and the possibility of multilayer stacks to further enhance the coupling toward deep-strong-coupling (g/ω > 1). The multimode behavior (odd vs even FP modes) provides additional control and must be considered when engineering spectra. The work also suggests that vacuum energy landscapes could be engineered laterally by patterning nanoparticle densities and potentially introducing chirality to realize chiral vacuum states. A complete accounting of all in-plane k modes would require regularization analogous to Casimir theory to obtain finite vacuum energy per unit area.

The study demonstrates room-temperature ultrastrong coupling between a Fabry-Pérot microcavity and a plasmonic nanorod array, achieving normalized coupling strengths up to gc/ωpl ≈ 0.55 with only ~4% cavity mode filling. Full Hopfield Hamiltonian analysis (including A^2 and counter-rotating terms) accurately models the spectra, in contrast to simplified models. Key USC signatures—large Rabi splittings, ground-state energy modification (~10%), and finite photonic occupancy (~0.06)—are observed or inferred. The platform opens pathways to explore USC physics in optical/IR regimes at ambient conditions and suggests routes to deep-strong coupling by stacking multiple nanoparticle layers or increasing oscillator strength. Future work may include multilayer arrays to achieve g/ωpl > 1, systematic optimization of rod length and density to maximize coupling, direct measurements of vacuum energy effects (e.g., forces on nanoparticles), exploiting engineered vacuum energy gradients, and implementing chiral nanoparticle arrays to realize chiral vacuum states.

• Modeling simplifications: Main analysis uses a single-mode Hopfield Hamiltonian; while justified by detuning of higher odd modes, full multimode effects and Fano-type interferences can shift extrema relative to true eigenenergies. A complete k-space treatment and polarization dependence are not exhaustively modeled.
• Measurement range: For the thinnest (100 nm) cavities, the upper polariton lay beyond the FTIR detection range and required separate visible-range measurements, introducing uncertainties (UP energy extraction error up to ±0.1 eV).
• Thin-cavity regime: The observed lower polariton back-bending in very thin cavities (<~200 nm) is likely due to near-field interactions with mirrors and is not captured by the Hamiltonian model; this regime is not analyzed in detail.
• Linewidth theory: A rigorous quantitative treatment of polariton linewidths in USC is not provided; master-equation approaches would be needed for full accuracy.
• Ground-state energy estimation: Indirect inference from reflectivity minima carries compounded errors, and the full vacuum energy per unit area (integration over k) diverges without regularization; such a treatment is left for future work.
• Structural variations: Slight increases in hybrid cavity thickness due to the array can shift bare FP modes (e.g., redshifts in 2nd FP mode), adding systematic uncertainty to detuning estimates.