  Recent microwave reflection measurements of Josephson junction ladders have
suggested the presence of nearly coherent collective charge oscillations deep
in the insulating phase. Here we develop a qualitative understanding of such
coherent charge modes by studying the local dynamical conductivity of the
insulating phase of a finite length sine-Gordon model. By considering
parameters near the non-interacting Fermion limit where the charge operator
dominantly couples to soliton-antisoliton pairs of the sine-Gordon model, we
find that the local dynamical resistance shows an array of sharp peaks in
frequency representing coherent phase oscillations on top of an incoherent
background. The strength of the coherent peaks relative to the incoherent
background increases as a powerlaw in frequency as well as exponentially as the
Luttinger parameter approaches a critical value. The dynamical conductivity
also clearly shows the insulating gap. We then compare the results in the high
frequency limit to a perturbative estimate of phase-slip-induced decay of
plasmons in the Josephson junction ladder.
