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 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.
ABSTRACT
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)/
ABSTRACT
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.
