A recent microscopic quantum theory of surface plasmon polaritons (Maurer, Allard, Todorov, Weick and Hagenmüller, 2026) quantizes the electronic polarization of a metal on Laplace-mode eigenbases and couples it nonperturbatively to the free-photon continuum, recovering localized surface plasmons of nanoparticles and the exact planar-interface surface-plasmon-polariton dispersion, but it is restricted to lossless metals, two benchmark geometries, and an unclarified relation to minimal-coupling formulations. The open problem, stated explicitly in that work, is to complete this framework: incorporate Ohmic and general dissipative channels consistently with Maxwell's macroscopic response, establish the precise relation between the truncated Power-Zienau-Woolley and minimal-coupling Hamiltonians, generalize the quantization to layered and coupled geometries, treat open quantum dynamics of emitters at arbitrary coupling strength, and derive the measurable quantum-optical consequences of the intrinsic ultrastrong light-matter coupling and ground-state fluctuations that the theory predicts.

Background

Surface plasmon polaritons (SPPs) are hybrid excitations of the electromagnetic field and collective free-electron oscillations at a metal-dielectric interface; they exist as propagating modes at planar interfaces (PSPs) and as localized surface plasmons (LSPs) confined to metallic nanoparticles. Because metals are strongly dispersive and absorptive, quantizing the electromagnetic field in their presence is a longstanding problem of quantum electrodynamics (QED). The standard resolution is reservoir quantization (Huttner and Barnett, 1992), in which the material polarization is coupled to a bath of harmonic oscillators that carries the loss; the macroscopic QED framework built on this idea (Scheel and Buhmann, 2008) expresses everything through the classical electromagnetic Green tensor. Parallel microscopic approaches have quantized SPPs in planar and spherical geometries within the minimal-coupling representation (Alpeggiani and Andreani, 2014) — where the electron-photon coupling enters through the vector potential and an A2\mathbf{A}^{2} diamagnetic term whose complicated form reflects broken translational symmetry — or via dipolar QED of the three-dimensional electron gas (Todorov, 2014). In the minimal-coupling treatment of nanoparticle chains the A2\mathbf{A}^{2} term was even neglected (Allard and Weick, 2021), making the mutual consistency of these representations unclear. The Power-Zienau-Woolley (PZW) representation of QED is obtained from the minimal-coupling (Coulomb-gauge) Hamiltonian by a unitary transformation that exchanges the coupling of charges to the vector potential A\mathbf{A} for a coupling of the polarization field P\mathbf{P} to the displacement field D\mathbf{D}; it is often advantageous for polaritonic systems because it separates electronic and photonic degrees of freedom optimally. Maurer et al. (2026) used it to build a microscopic quantum theory of SPPs valid, in principle, for arbitrary metal-dielectric geometries: the polarization field is expanded on discrete modes obtained by solving the Laplace equation for the geometry, and the fundamental electronic oscillator is the bulk plasmon (the collective oscillation of the metal's free-electron gas at the plasma frequency ωp\omega_{\mathrm{p}}), coupled nonperturbatively to the continuum of free radiation modes. The theory recovers LSPs including radiative frequency shifts and decay, the exact PSP dispersion at a planar interface, and predicts that light-matter interaction at metal-dielectric interfaces is intrinsically in the ultrastrong-coupling (USC) regime — where the interaction strength is a sizable fraction of the bare transition frequencies (Forn-Díaz et al., 2019; Frisk Kockum et al., 2019) — so that the polaritonic ground state contains a finite bulk-plasmon population and exhibits anomalous quantum fluctuations controllable by the refractive index. However, the published theory neglects nonradiative Ohmic losses, treats only the sphere and the planar interface, and truncates the electronic basis (dipolar Laplace modes for the nanoparticle; only exponentially decaying solutions of Laplace's equation for the interface). Under such truncations the exact unitary PZW transformation no longer connects the resulting Hamiltonian to its minimal-coupling counterpart, so the equivalence of representations — a general problem sharpened by Stokes and Nazir (2023), who showed that material truncation generically breaks gauge invariance in USC QED, and by Savasta et al. (2021) for two-level truncations — is left unresolved for these specific plasmonic mode bases. Related emitter-level questions are also open: whether USC can arise from coupling to transverse photons at all has been challenged by Fernández de la Pradilla, Moreno and Feist (2025), who argue it originates in Coulomb interactions instead.

Problem Statement

The problem, as stated in the concluding outlook of Maurer, Allard, Todorov, Weick and Hagenmüller (2026), is to complete their microscopic Power-Zienau-Woolley (PZW) quantum theory of surface plasmon polaritons — which is currently lossless, verified only for a metallic nanosphere and a single planar interface, and of unclarified relation to other representations — into a controlled and experimentally testable quantum theory of plasmonic systems. A complete solution must deliver all five of the following jointly linked results (addressing a proper subset resolves only the corresponding components and does not resolve the task): (i) incorporate nonradiative Ohmic losses (and general dissipative channels) into the microscopic PZW Hamiltonian by coupling the electronic degrees of freedom to bath oscillators, for example with an Ohmic spectral density J(ω)∝γωJ(\omega)\propto\gamma\omega, in a way consistent with Maxwell's equations and the macroscopic dielectric response, and reducing exactly to the published lossless Hamiltonian in the zero-damping limit; (ii) establish the precise relation between the PZW and minimal-coupling Hamiltonians for the practically truncated mode bases used for plasmonic systems (dipolar Laplace modes for nanoparticles, evanescent-sector modes for interfaces), characterizing when and how the unitary PZW transformation fails and what the incomplete-basis inequivalence implies for physical predictions such as the planar SPP dispersion both representations nevertheless reproduce; (iii) generalize the Laplace-mode quantization and the nonperturbative diagonalization to layered geometries and other structured or coupled systems (e.g., coupled nanoparticles and parallel metallic plates), where propagating and near-field couplings compete, and recover the corresponding classical results such as thin-film surface-plasmon dispersions; (iv) develop controlled open-quantum-system treatments — e.g., Fano diagonalization (the exact diagonalization of discrete matter modes hybridized with the free-photon continuum) or Green's-function bath embeddings — of the interaction of the quantized SPP modes with quantum emitters — single two-level systems or ensembles — at arbitrary coupling strength, yielding observable emission spectra; and (v) determine how the geometry-controlled ground-state plasmon populations and anomalous quantum fluctuations predicted by the intrinsic ultrastrong coupling translate into at least one measurable quantum-optical phenomenon — nonclassical light emission, super- and subradiant collective states (collective states of emitter ensembles whose radiative decay is, respectively, enhanced or suppressed by constructive or destructive interference), or Casimir–Polder interactions (the retarded van der Waals potential between a quantum emitter and a nearby surface, mediated by the quantized electromagnetic field) — in experimentally realistic parameter regimes. Each result (i)–(v) counts as delivered when it supplies a single explicit, self-consistent construction, derivation, or quantitative prediction of the kind it specifies; in particular, where a result leaves open which dissipative channels or which geometries to treat, one representative worked case suffices to deliver that part, with the remaining members of the class as optional extensions. A complete solution must make the common objective explicit: a single, internally consistent microscopic quantum framework in which dissipation, gauge consistency, geometry, and emitter coupling are treated on the same footing as the published lossless theory, and whose predictions can be confronted with experiments.

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