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

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 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 two-dimensional (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.

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)> greater, similar 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.

Phase locking in Josephson ladders and the discrete sine-Gordon equation: the effects of boundary conditions and current-induced magnetic fields

We report on the stability of phase-locked solutions to ladder arrays of underdamped Josephson junctions under both periodic and open boundary conditions and in the presence of current-induced magnetic fields. We calculate the Floquet exponents based on the resistively and capacitively shunted junction (RCSJ) model, as well as on a simplified model of the ladder that leads to a discrete sine-Gordon (DSG) equation for the horizontal, current-biased junctions. In the case of zero induced magnetic fields, we find the DSG equation (commonly applied to parallel arrays) appreciably overestimates the exponents of the full ladder in the overdamped regime (corresponding to the limit of small junction capacitance, beta(c)), and that difference physically results from differing spectra for small-amplitude phase oscillations of the DSG and RCSJ equations. mutual inductance between plaquettes is included we find there are ranges of values for the mutual inductance for which the ladder is in fact unstable. To understand the cause of the observed instabilities, it is crucial to consider the behavior of the vertical junctions.

Floquet exponents of underdamped josephson ladders: A comparison with predictions of the discrete sine-gordon equation

We calculate Floquet exponents for phase-locked solutions in ladder arrays of Josephson junctions in zero external field. We assume a resistively and capacitively shunted junction (RCSJ) model, and we allow for critical current anisotropy between the horizontal and vertical junctions. The ladders range in size from 5 to 30 plaquettes and are biased along the rungs with uniform dc bias currents. The Floquet exponents quantify the stability of the solutions and are calculated numerically for the RCSJ model as a function of junction capacitance (beta(c)) as well as critical current anisotropy (Lambda). We also model the array with the discrete sine-Gordon (DSG) equation, and we are able to calculate the exponents analytically in that case. We find the analytic results from the DSG equation agree quantitatively with the numerical results from the RCSJ model over a wide range of beta(c) and Lambda values and even agree qualitatively for beta(c)-->1 and Lambda-->0. Based on the analytic result we argue that perturbations in the array are damped by the small-angle phase oscillations of the underlying lattice (the "phonons" of the lattice), and like a classical harmonic oscillator with damping, each phonon mode has a crossover (as a function of decreasing beta(c) or Lambda) from underdamped to overdamped dynamics. Such crossover behavior is clearly visible in the results for the Floquet exponents and is manifested as a maximum in the Floquet exponent as a function of the junction capacitance. This intriguing result speaks to the opportunity, in principle, of tuning the capacitance such as to optimize the stability of the phase-locked solutions.