Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model

We show that the resistively shunted junction (RSJ) equations describing a ladder array of overdamped, critical-current disordered Josephson junctions that are current biased along the rungs of the ladder can be mapped onto a Kuramoto model with nearest neighbor, sinusoidal couplings. This result is obtained by an averaging method, in which the fast dynamics of the RSJ equations are integrated out, leaving the dynamics which describe the time scale over which neighboring junctions along the rungs of the ladder phase and frequency synchronize. We quantify the degree of frequency synchronization of the rung junctions by calculating the standard deviation of their time-averaged voltages, ${\sigma }_{\omega },$ and the phase synchronization is quantified by calculating the time average of the modulus of the Kuramoto order parameter, $〈|r|〉.$ We test the results of our averaging process by comparing the values of ${\sigma }_{\omega }$ and $〈|r|〉$ for the original RSJ equations and our averaged equations. We find excellent agreement for dc bias currents of $I_B/〈I_c〉\gtrsim 3,$ where $〈I_c〉$ is the average critical current of the rung junctions, and critical current disorders of up to 10%. We also study the effects of thermal noise on the synchronization properties of the overdamped ladder. Finally, we find that including the effects of junction capacitance can lead to a discontinuous synchronization transition as the strength of the coupling between neighboring junctions is smoothly varied.