  Josephson junctions are the most prominent nondissipative and at the same
time nonlinear elements in superconducting circuits allowing Cooper pairs to
tunnel coherently between two superconductors separated by a tunneling barrier.
Due to this, physical systems involving Josephson junctions show highly complex
behavior and interesting novel phenomena. Here, we consider an infinite
one-dimensional chain of superconducting islands where neighboring islands are
coupled by capacitances. We study the effect of Josephson junctions shunting
each island to a common ground superconductor. We treat the system in the
regime where the Josephson energy exceeds the capacitive coupling between the
islands. For the case of two offset charges on two distinct islands, we
calculate the interaction energy of these charges mediated by quantum phase
slips due to the Josephson nonlinearities. We treat the phase slips in an
instanton approximation and map the problem onto a classical partition function
of interacting particles. Using the Mayer cluster expansion, we find that the
interaction potential of the offset charges decays with an universal
inverse-square power law behavior.
