Quantum statistical mechanics of an array of resistively shunted Josephson junctions

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, $E_0$≫V≫$k_B$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.