  We consider a chain of regularly-spaced spherical metallic nanoparticles,
where each particle supports three degenerate localized surface plasmons. Due
to the dipolar interaction between the nanoparticles, the localized plasmons
couple to form extended collective modes. Using an open quantum system approach
in which the collective plasmons are interacting with vacuum electromagnetic
modes and which, importantly, readily incorporates retardation via the
light-matter coupling, we analytically evaluate the resulting radiative
frequency shifts of the plasmonic bandstructure. For subwavelength-sized
nanoparticles, our analytical treatment provides an excellent quantitative
agreement with the results stemming from laborious numerical calculations based
on fully-retarded solutions to Maxwell's equations. Indeed, the explicit
expressions for the plasmonic spectrum which we provide showcase how including
retardation gives rise to a logarithmic singularity in the bandstructure of
transverse-polarized plasmons. We further study the impact of retardation
effects on the propagation of plasmonic excitations along the chain. While for
the longitudinal modes, retardation has a negligible effect, we find that the
retarded dipolar interaction can significantly modify the plasmon propagation
in the case of transverse-polarized modes. Moreover, our results elucidate the
analogy between radiative effects in nanoplasmonic systems and the cooperative
Lamb shift in atomic physics.
