Quantum Fluctuations in the Equilibrium State of a Thin Superconducting   Loop
  We study the oscillatory flux dependence of the supercurrent in a thin superconducting loop, closed by a Josephson junction. Quantum fluctuations of the order parameter in the loop affect the shape and renormalize the amplitude of the supercurrent oscillations. In a short loop, the amplitude of the sinusoidal flux dependence is suppressed. In a large loop, the supercurrent shows a saw-tooth dependence on flux in the classical limit. Quantum fluctuations not only suppress the amplitude of the oscillations, but also smear the cusps of the saw-tooth dependence. The oscillations approach a sinusoidal form with increasing fluctuation strength. At any finite length of the loop, the renormalized current amplitude is finite. This amplitude shows a power-law dependence on the junction conductance, with an exponent depending on the low-frequency impedance of the loop. 
F. W. J. Hekking and L. I. Glazman