Resonant-cavity-induced phase locking and voltage steps in a Josephson array

We describe a simple dynamical model for an underdamped Josephson junction array coupled to a resonant cavity. From numerical solutions of the model in one dimension, we find that (i) current-voltage characteristics of the array have self-induced resonant steps (SIRS), (ii) at fixed disorder and coupling strength, the array locks into a coherent, periodic state above a critical number of active Josephson junctions, and (iii) when $N_a$ active junctions are synchronized on an SIRS, the energy emitted into the resonant cavity is quadratic with $N_a.$ All three features are in agreement with a recent experiment [P. Barbara, A. B. Cawthorne, S. V. Shitov, and C. J. Lobb, Phys. Rev. Lett. 82, 1963 (1999)].