Microscopic quantum description of surface plasmon polaritons: revealing intrinsic ultrastrong light-matter coupling

TL;DR

This work presents a microscopic quantum description of surface plasmon polaritons at arbitrary metal–dielectric interfaces using the Power-Zienau-Woolley representation, with the bulk plasmon as the fundamental electronic oscillator nonperturbatively coupled to radiative photon modes. The authors derive a complete Hamiltonian framework, quantize the polarization field via Laplace-based modes, and show that surface plasmon frequencies arise from geometry through $\omega_\mu = \omega_p/\sqrt{1+\lambda_\mu}$. Applying the theory to both spherical nanoparticles and planar interfaces recovers known LSP and PSP spectra, while uncovering intrinsic ultrastrong light–matter coupling evidenced by finite ground-state plasmon populations and nonperturbative renormalizations. The results provide a rigorous quantum plasmonics toolkit, with implications for quantum emitters, Casimir-type interactions, and layered nanophotonic devices, and suggest avenues for incorporating losses and extending to more complex geometries.

Abstract

We develop a microscopic quantum theory of surface plasmon polaritons valid for arbitrary metal-dielectric geometries. Our framework is based on the Power-Zienau-Woolley representation of quantum electrodynamics, which provides an optimal separation between electronic and photonic degrees of freedom and is therefore particularly well suited for constructing quantum descriptions of polaritonic excitations in strongly dispersive media. Within this formulation, the fundamental electronic oscillator is identified as the bulk plasmon mode, which is nonperturbatively coupled to the radiative continuum of free photon modes. This coupling induces a geometry-dependent renormalization of the bulk plasma frequency, giving rise to confined plasmonic resonances. As specific applications, we recover the localized surface plasmon modes of metallic nanoparticles, including radiative frequency shifts and decay, as well as the exact dispersion relation of propagating surface plasmon polaritons at planar interfaces. Our quantum treatment further reveals that light-matter interactions at metal-dielectric interfaces are inherently in the ultrastrong coupling regime. As a result, in the quasistatic limit, the system exhibits unconventional ground-state quantum fluctuations that can be controlled through the refractive index. These results open new intriguing perspectives in the field of quantum plasmonics.

Figures (9)

• Figure 1: Schematic representation of a metal-dielectric interface. The system comprises a metallic region (1) of volume $V$, assumed to exhibit negligible nonradiative losses and described by a Drude-type dielectric function $\varepsilon_{1}(\omega)=\varepsilon_{\infty}- \omega^{2}_{\mathrm{p}}/\omega^2$, where $\varepsilon_{\infty}$ denotes the background dielectric constant, and $\omega_{\mathrm{p}}$ is the plasma frequency. Adjacent to this is a dielectric region (2), characterized by a positive dielectric constant $\varepsilon_2 >0$. The unit vector $\mathbf{N}$ is oriented perpendicular to the interface, pointing into the dielectric medium.
• Figure 2: Spherical metallic nanoparticle of radius $a$ with a dielectric function $\varepsilon_1(\omega)$ [cf. Eq. \ref{['eq : Drude']}], surrounded by a dielectric medium of dielectric constant $\varepsilon_2$. Illustrations of the longitudinal (solid blue lines) and transverse (red dotted lines) polarization fields for the fundamental localized plasmon mode ($l = 1$) of the nanoparticle are represented. The longitudinal and transverse polarizations are constant inside the sphere and oriented along the unit vector $\hat{\mathbf{z}}$. The field lines of $\mathbf{P}_{\perp}$ form closed loops, whereas those of $\mathbf{P}_{\parallel}$ originate at the negative charges and terminate at the positive charges.
• Figure 3: Ground-state population of bulk plasmons for a metallic sphere (red dots) and its complementary spherical cavity (blue squares) as a function of the mode order $l$ for $\varepsilon_2 = 1$ and $\varepsilon_{\infty} = 1$. The dotted line corresponds to the asymptotic limit of a flat interface.
• Figure 4: Planar interface separating a metal with dielectric function $\varepsilon_1(\omega)$ for $z<0$ and a dielectric medium of constant $\varepsilon_2$ for $z>0$. The field lines of longitudinal polarization (blue, solid) and transverse polarization (red, dotted) are shown. As displayed in Fig. \ref{['fig : sphere']}, the field lines of $\mathbf{P}_{\perp}$ are closed loops and those of $\mathbf{P}_{\parallel}$ start at the negative charges and end at the positive charges.
• Figure 5: (a) PSP polariton dispersion $\Omega_{k_{\parallel}}$ from Eq. \ref{['eq : Disp SPP']} (purple). The light line $ck_{\parallel}$ is in red (dotted), and the surface plasmon frequency $\omega_\mathrm{sp}$ is shown as a blue dashed line. (b) Electronic (blue, solid) and photonic (red, dash-dotted) weights of PSP polaritons, $\eta_{\mathbf{k}_{\parallel}}^{\mathrm{el}}$ and $\eta_{\mathbf{k}_{\parallel}}^{\mathrm{ph}}$. (c) Square modulus of the anomalous Hopfield coefficient, $\vert x_{\mathbf{k}_{\parallel}} \vert^2$. The dotted line represents the asymtptotic value in the quasistatic limit, $k_{\parallel} \gg \omega_{\mathrm{p}}/c$, which corresponds to the plasmon population in the ground state.