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 ͑RCSJ͒ 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 ͑2D͒ 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.

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)/ greater, similar 3, where 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.

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We show that a serial array of N nonuniform, underdamped Josephson junctions coupled piezoelectrically to a nanoelectromechanical ͑NEM͒ oscillator results in phase locking of the junctions. Our approach is based on a semiclassical solution to a set of coupled differential equations that were generated by the Heisenberg operator equations, which in turn are based on a model Hamiltonian that includes the following effects: the charging and Josephson energies of the junctions, dissipation in the junctions, the effect of a dc bias current, an undamped simple harmonic oscillator ͑representing the NEM͒, and an interaction energy ͑due to the piezoelectric effect͒ between the NEM and the junctions. Phase locking of the junctions is signaled by a step in the current-voltage ͑I-V͒ curve. We find the phase-locked states are ͑neutrally͒ stable at the bottom and top of the step but not for bias currents in the middle of the step. Using harmonic balance, we are able to calculate an analytical expression for the location of the resonance step, v step , in the I-V curve. Because of the multistability of the underdamped junctions, it is possible, with a judicious choice of initial conditions and bias current, to set a desired number N a ഛ N of junctions on the resonance step, with N − N a junctions in the zero-voltage state. We are also able to show that, when N a junctions are in the phase-locked configuration, the time-averaged energy of the NEM oscillator scales like N a 2 .

We report the results of a Floquet analysis of two-dimensional arrays of resistively and capacitively shunted Josephson junctions in an external transverse magnetic field. The Floquet analysis indicates stable phase locking of the active junctions over a finite range of values of the bias current and junction capacitance, even in the absence of an external load. This stable phase locking is robust against critical current disorder, up to at least a Ϯ25% rms spread in critical currents.