Synchronization in a one-dimensional array of point Josephson junctions coupled to a common load

We study the synchronization in a one-dimensional array of point Josephson junctions coupled to a common capacitor, which establishes a long-range interaction between junctions and synchronizes them. The stability diagram of synchronization in a noise-free system is obtained. The current when junctions transform from resistive state into zero-voltage state is then calculated and its dependence on the shunt parameters and the dissipation of junctions is revealed. In the presence of thermal noise, the synchronized oscillations are destroyed at a critical temperature and the system undergoes a continuous phase transition of desynchronization. A possible stability diagram of the synchronized oscillations with respect to thermal noise, current, dissipations, and shunt capacitance is then constructed. Finally we investigate the dynamic relaxation from random oscillations into a synchronized state. The relaxation time increases with the system size and temperature, but is reduced by the shunt capacitor.