Surface plasmaritons: Wave-mechanical and second-quantized theories

Abstract

In the wake of recently established quantum theories for bulk and surface plasmons, and for bulk plasmaritons [Jung and Keller, Phys. Rev. A 103, 063501 (2021); Phys. Rev. A 104, 053508 (2021)] wave-mechanical and second-quantized (QED) theories for surface plasmaritons are developed. Our photon wave-mechanical theory is based on the covariant four-potential in the Lorenz gauge, where dynamical equations for surface plasmariton quantum particles consisting of scalar ($S$), longitudinal ($L$), and transverse ($T1$, $T2$) photons and driven by a surface current density sheet are obtained. The second quantized QED theory of surface plasmaritons is formulated on the basis of the Heisenberg equations of motion for the contravariant four-potential annihilation operators. The connection of the surface plasmariton quantum theory to the quasiparticle picture of bulk plasmaritons is elucidated by studying the $T$-photon exponential decay lengths appearing in the description. The shortest decay length (equal on both sides of the surface) characterizes the one-dimensional spatial confinement of the $T$-photon source. Starting from a reexamination (and correction) of the dynamic boundary conditions for a flat jellium-vacuum interface carrying a surface current density, the dispersion relation for surface plasmaritons is obtained. The explicit form of the dispersion relation is derived upon a study of the local field in the selvedge region of the boundary. Fundamental aspects of the dispersion relation are pointed out, modeling the electron dynamics by the microscopic electrodynamics of a quantum well. Remarks on (i) our covariant surface plasmariton quantum theory seen in a broader perspective, (ii) suggestions for applications of the theory, and (iii) a comparison of the key ingredients in our trio of papers on plasmon and plasmariton quantum physics are presented.