One-dimensional Josephson arrays as superlattices for single Cooper pairs

We investigate uniform one-dimensional arrays of small Josephson junctions ($E_J \ll E_C$, $E_C = (2e)^2/2C$) with a realistic Coulomb interaction $U(x) = E_C \lambda \exp( - |x|/\lambda)$ (here $\lambda \gg 1$ is the screening length in units of the lattice constant of the array). At low energies this system can be described in terms of interacting Bose particles (extra single Cooper pairs) on the lattice. With increasing concentration $\nu$ of extra Cooper pairs, a crossover from the Bose gas phase to the Wigner crystal phase and then to the superlattice regime occurs. The phase diagram in the superlattice regime consists of commensurable insulating phases with $\nu = 1/l$ ($l$ is integer) separated by superconducting regions where the current is carried by excitations with {\em fractional} electric charge $q = \pm 2e/l$. The Josephson current through a ring-shaped array pierced by magnetic flux is calculated for all of the phases.