Quantum phase transition and possible phase separation in ultrathin doubly-connected superconducting cylinders of Al

Fluxoid quantization demands that the superfluid velocity, $v_s$, of a doubly-connected superconducting cylinder increase as its diameter, $d$, decreases, leading to a destructive regime and a quantum phase transition (QPT) in one dimension (1D) - Superconductivity is suppressed around half-integer flux quanta even at zero temperature for cylinders with a $d$ less than the zero-temperature superconducting coherence length, $\xi(0)$. We have fabricated ultrathin doubly-connected superconducting cylinders of Al over a wide range of $d/\xi (0)$ ratios, with the smallest cylinder down to 100 nm in diameter. Electrical transport measurements revealed the presence of robust step-like features in resistance $vs.$ temperature curves as the destructive regime is approached. These field-induced step-like features, present only in the smallest cylinders with $d/\xi (0) $ close to or less than 1, are unrelated to sample inhomogeneity or phase slip centers, and are most likely resulted from a phase separation close to the QPT. We have also found that the normal state in the destructive regime becomes less stable as the $d/\xi (0)$ ratio increases, with the resistances at the half-flux quantum suppressed continuously from its full normal-state value. Resistance $vs.$ magnetic field measurements show that the field tuned QPT is very sharp (less than 1G) with little evidence for hysteretic behavior.