Length-Scale Dependence of the Superconductor-to-Insulator Quantum Phase Transition in One Dimension

One-dimensional (1D) arrays of small-capacitance Josephson junctions demonstrate a sharp transition, from Josephson-like behavior to the Coulomb blockade of Cooper-pair tunneling, as the effective Josephson coupling between nearest neighbors is tuned with an externally applied magnetic field. Comparing the zero-bias resistance of three arrays with 255, 127, and 63 junctions, we observe a critical behavior where the resistance, extrapolated to $T=0\mathrm{}$, is independent of length at a critical magnetic field. Comparison is made with a theory of this $T=0\mathrm{}$ quantum phase transition, which maps to the 2D classical $\mathrm{XY}$ model.