Theory of phase locking in simple two-dimensional Josephson junction arrays with small inductances

Several elementary two-dimensional Josephson junction arrays have been investigated mainly analytically with the special emphasis on phase locking. For this purpose, a novel perturbation scheme appropriate for the handling of small loop inductances has been developed and applied to (i) a single SQUID cell, (ii) two cells coupled by an inductive line, and (iii) an externally loaded two-cell configuration with more general internal shunt. While internal in-phase locking within the cells can easily be achieved as long as the normalized ring inductances are small compared to unity, the inter-cell coupling tends to favor an antiphase regime. A mechanism is proposed for circumventing this problem and the boundaries separating both regimes are determined. The influence of an external load is considered as well.