  Even in the absence of Coulomb interactions phase fluctuations induced by
quantum size effects become increasingly important in superconducting
nano-structures as the mean level spacing becomes comparable with the bulk
superconducting gap. Here we study the role of these fluctuations, termed
"quantum capacitance", in the phase diagram of a one-dimensional (1D) ring of
ultrasmall Josephson junctions (JJ) at zero temperature by using path integral
techniques. Our analysis also includes dissipation due to quasiparticle
tunneling and Coulomb interactions through a finite mutual and self
capacitance. The resulting phase diagram has several interesting features: A
finite quantum capacitance can stabilize superconductivity even in the limit of
only a finite mutual-capacitance energy which classically leads to breaking of
phase coherence. In the case of vanishing charging effects, relevant in cold
atom settings where Coulomb interactions are absent, we show analytically that
superfluidity is robust to small quantum finite-size fluctuations and identify
the minimum grain size for phase coherence to exist in the array. We have also
found that the renormalization group results are in some cases very sensitive
to relatively small changes of the instanton fugacity. For instance, a certain
combination of capacitances could lead to a non-monotonic dependence of the
superconductor-insulator transition on the Josephson coupling.
