Synchronization in disordered Josephson junction arrays: Small-world connections and the Kuramoto model

TL;DR

This paper investigates how small-world, long-range connections influence synchronization in disordered Josephson junction arrays, using phase models derived from the RCSJ equations, revealing that shortcuts facilitate synchronization in 1D but are less effective in 2D.

Contribution

It derives a second order phase model from the RCSJ equations for ladder arrays with capacitance and analyzes the impact of small-world shortcuts on synchronization in 1D and 2D arrays.

Findings

Shortcuts enhance synchronization in 1D ladder arrays.

Shortcuts have limited effect on 2D array synchronization.

Effective phase models explain differences between 1D and 2D behaviors.

Abstract

We study synchronization in disordered arrays of Josephson junctions. In the first half of the paper, we consider the relation between the coupled resistively- and capacitively shunted junction (RCSJ) equations for such arrays and effective phase models of the Winfree type. We describe a multiple-time scale analysis of the RCSJ equations for a ladder array of junctions \textit{with non-negligible capacitance} in which we arrive at a second order phase model that captures well the synchronization physics of the RCSJ equations for that geometry. In the second half of the paper, motivated by recent work on small world networks, we study the effect on synchronization of random, long-range connections between pairs of junctions. We consider the effects of such shortcuts on ladder arrays, finding that the shortcuts make it easier for the array of junctions in the nonzero voltage state to synchronize. In 2D arrays we find that the additional shortcut junctions are only marginally effective at inducing synchronization of the active junctions. The differences in the effects of shortcut junctions in 1D and 2D can be partly understood in terms of an effective phase model.