Quantum statistical mechanics of an array of resistively shunted Josephson junctions.

TLDR: A fully quantum-mechanical model of an ordered array of resistively shunted Josephson junctions is constructed, and it is found that in the extreme quantum limit, the onset of global phase coherence (superconductivity) in the array occurs only if R is less than a critical value.

Abstract: We have constructed a fully quantum-mechanical model of an ordered array of resistively shunted Josephson junctions, and have determined the nature of the phase diagram as a function of the Josephson coupling, V, the capacitance, C (or, equivalently, the charging energy ${E}_{0}$=4${e}^{2}$/C), the shunt resistance, R, and the temperature, T. In order to treat the dissipative element (R) in a quantum system, we have modeled it by a heat bath with spectral weight chosen to reproduce Ohmic resistance in the classical limit. Among other results, we find that in the extreme quantum limit, ${E}_{0}$\ensuremath{\gg}V\ensuremath{\gg}${k}_{B}$T, the onset of global phase coherence (superconductivity) in the array occurs only if R is less than a critical value ${R}_{c}$=Ah/${e}^{2}$, where A is a number of order 1 which depends on the dimension and the lattice structure. The fact that the dissipation enters the thermodynamics at all is a consequence of the quantum nature of the transition. This transition is reminiscent of the results of recent experiments on thin films of granular superconductors.