Experiments on single surface plasmon polaritons (SPPs) propagating in metallic stripe waveguides at 808 nm show that loss acts as a linear, uncorrelated Markovian amplitude-damping channel: the normalized quantum statistics, such as the second-order coherence g(2)(0)g^{(2)}(0), are invariant under propagation while the mean intensity decays exponentially. The source work (Di Martino et al., 2012) explicitly flags the regime closer to the plasmon resonance as uninvestigated: there the SPP acquires a more electron-like character, and correlated microscopic damping channels — excited phonons, background ion cores, electron-gas collisions, and interband transitions creating electron–hole pairs — may make the damping depart from the linear uncorrelated model. The open problem is to determine whether the quantum decoherence of propagating SPPs in this near-resonance regime still follows the linear uncorrelated Markovian model or departs from it because of correlations between and within the microscopic damping channels, using quantum-statistics measurements (g(2)g^{(2)}, Fock-state populations, amplitude- and phase-damping times) close enough to resonance to test it. All existing quantum-regime measurements operate far from resonance, and existing non-Markovian plasmonic theory addresses emitters coupled to SPP modes rather than the damping of the SPP quantum state itself.

Background

A surface plasmon polariton (SPP) is a hybrid excitation at a metal–dielectric interface in which an electromagnetic wave is coupled to a collective oscillation of the conduction-electron charge density; it can be guided along lithographically defined metal stripes at subwavelength confinement. In quantum-plasmonics experiments, single photons from spontaneous parametric down-conversion (SPDC, a χ(2)\chi^{(2)} nonlinear process producing correlated photon pairs in a nonlinear crystal such as beta barium borate (BBO)) are grating-coupled into an SPP waveguide, and the light out-coupled at the far end is analyzed with a Hanbury Brown–Twiss setup (a beamsplitter plus two single-photon detectors operated in coincidence) to measure the second-order coherence g(2)(τ)=⟨a^†a^†a^a^⟩/⟨a^†a^⟩2g^{(2)}(\tau)=\langle \hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}\rangle/\langle \hat{a}^{\dagger}\hat{a}\rangle^{2}, whose value at zero delay distinguishes thermal (g(2)(0)=2g^{(2)}(0)=2), coherent (=1=1), and single-photon (<1<1) fields, and to reconstruct Fock-state populations (the probabilities PnP_n of occupying the nn-photon number states ∣n⟩|n\rangle). Di Martino et al. (Nano Lett. 12, 2504 (2012)) studied 3-μ\mum-wide, 150-nm-thick gold stripes at a free-space wavelength of 808 nm. Because the metal permittivity there is well described by a free-electron (Drude) response and the SPP is far from the plasmon resonance (the short-wavelength cutoff of the SPP dispersion, near which the SPP wavelength shrinks toward the metal's screened-plasma wavelength and the excitation becomes predominantly an electron-density oscillation, overlapping the interband dd-band-to-Fermi-surface transitions of gold near 500–550 nm), they modeled propagation loss as a linear, uncorrelated Markovian environment: a bosonic amplitude-damping channel that rescales mean photon numbers by an efficiency η\eta while leaving g(2)(0)g^{(2)}(0) unchanged. Their data — mean excitation rates, g(2)(0)g^{(2)}(0) constant over waveguide lengths from 5 to 30 μ\mum, and Fock-state populations — were consistent with this model, evidencing that single SPPs behave as well-defined bosonic modes with classical-like exponential attenuation. (Their measured propagation length, ℓ=8.9±1.7\ell=8.9\pm1.7 μ\mum, fell short of a finite-element-method (FEM) electromagnetic simulation prediction of 16.7 μ\mum, attributed to fabrication imperfections — an engineering discrepancy rather than the physics question here.) They explicitly anticipated that this picture could fail: at the single-excitation level, correlations between and within the microscopic damping channels (excited phonons, background ion cores, electron-gas collisions, and interband electron–hole pair production) could make the damping depart from the classical model, most plausibly closer to the plasmon resonance where the SPP character becomes more electron-like; in their words, this regime "remains to be investigated". Later work sharpened the Markovian picture without entering that regime: the same platform demonstrated plasmonic Hong–Ou–Mandel (two-photon) interference (Di Martino et al., 2014), two-plasmon interference at photon-level visibility ruled out detectable single-plasmon nonlinearities (Fakonas et al., 2014), and a Mach–Zehnder-type measurement decomposed single-SPP decoherence into an amplitude-damping time T1T_1 and phase-damping times T2,T2∗T_2, T_2^{*} at 810 nm, finding amplitude damping dominant with non-negligible pure dephasing (Dlamini et al., 2018) — a phenomenological channel split, still not a microscopic, channel-resolved theory, and still far from resonance. Separately, it has been shown that the "quantum statistics are preserved" expectation is not universal: multiparticle plasmonic subsystems can exhibit modified quantum statistics governed by bosonic or even fermionic coherence processes (You et al., 2021; Hong et al., 2024), and non-Markovian open-system treatments have been developed for emitters coupled to SPP waveguide modes (Yang et al., 2024). None of these addresses the damping of the SPP quantum state itself near resonance.

Problem Statement

For a single (or few) surface plasmon polariton propagating along a real metallic waveguide at wavelengths approaching the metal's plasmon resonance — the regime where the free-electron approximation fails, the SPP acquires significant electron-like character, and interband electron–hole creation, electron–phonon, and electron–ion-core/electron-gas damping channels are simultaneously active — determine whether the quantum-state evolution is still described by a linear, uncorrelated Markovian amplitude-damping channel (under which the mean intensity decays exponentially and normalized quantum statistics such as g(2)(0)g^{(2)}(0) and the loss-corrected (efficiency-normalized) Fock-state populations are invariant), or whether correlations between and within the microscopic damping channels produce observable departures such as excess pure dephasing beyond the far-from-resonance baseline T2∗T_2^{*} measured at 810 nm, non-exponential attenuation, non-Markovian memory, or a change of g(2)(0)g^{(2)}(0)/loss-corrected Fock-state populations with propagation length. The central task is to determine which of these alternatives holds — either by an executed quantum-statistics measurement (second-order coherence, Fock-state populations, and/or T1T_1, T2T_2, T2∗T_2^{*} decomposition) on guided SPPs operated close enough to the plasmon resonance to probe the electron-like regime, or by a detailed experiment-plus-theory analysis of that regime that establishes the outcome such a measurement would give. Both forms resolve the task; a measurement proposal that leaves the outcome unestablished does not. A channel-resolved microscopic description of the damping — for instance a master equation (or equivalent open-system description) for a quantized SPP that resolves the individual damping channels from the metal's electronic structure (Drude plus interband response, phonon and collision damping) and includes their inter- and intra-channel correlations — would strengthen the interpretation of any observed departure, but it accompanies the question rather than being imposed as a completion requirement: the source's open question is whether the regime closer to the plasmon resonance departs from the linear uncorrelated model, and a rigorously established 'no observable departure' is an admissible answer. The overarching scientific goal is to establish, on quantum-statistics evidence, whether the linear uncorrelated Markovian loss model used throughout quantum plasmonics remains valid for guided SPPs in the near-resonance regime, replacing or confirming it.

References

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