  We consider a system of weakly interacting bosons confined on a planar double
lattice ring subjected to two artificial gauge fields. This system is known to
display three phases, the Meissner phase where the flow of particles is carried
at the edges of the system without transverse current, a vortex phase
characterized by non-zero transverse current, and a biased-ladder phase,
characterized by an imbalance of the population of the two rings. We use the
Bogoliubov approximation to determine the excitation spectrum in the three
phases, the dynamic structure factor and the quantum fluctuation corrections to
the first-order correlation function. Our analysis reveals supersolid features
as well as Josephson modes, corresponding to out-of-phase modes of the finite
ring.
