Quantum Theory of Surface Plasmon Polaritons: Planar and Spherical Geometries

Abstract A quantum theory of retarded surface plas-mons on a metal–vacuum interface is formulated, by analogy with the well-known and widely exploited the-ory of exciton-polaritons. The Hamiltonian for mutually interacting instantaneous surface plasmons and transverse electromagnetic modes is diagonalized with recourse to a Hopfield–Bogoljubov transformation, in order to obtain a new family of modes, to be identified with retarded plas-mons. The interaction with nearby dipolar emitters is treated with a full quantum formalism based on a general defi-nition of modal effective volumes. The illustrative cases of a planar surface and of a spherical nanoparticle are considered in detail. In the ideal situation of absence of dissipation, as an effect of the conservation of in-plane wavevector, retarded plasmons on a planar surface repre-sent true stationary states (which are usually called surface plasmon polaritons), whereas retarded plasmons in a spher-ical nanoparticle, characterized by frequencies that overlap with the transverse electromagnetic continuum, become res-onances with a finite radiative broadening. The theory presented constitutes a suitable full quantum framework for the study of nonperturbative and nonlinear effects in plasmonic nanosystems