Phase-locked oscillator optimization for arrays of Josephson junctions

Abstract

Au overview of phase locking in two-dimensional (2D) arrays of identical Josephson junctions is presented. General design criteria are discussed for optimization of power and linewidth. A harmonic balance technique is used to derive an analytic expression for the fundamental power as a function of bias voltage for a single shunted tunnel junction with an external shunt resistor having parasitic inductance. A linear stability analysis is performed on the in-phase state of 2D arrays in the absence of any external load. Most excitation modes in the 2D array are damped, leading to stable phase locking between parallel junctions within each row; however, within the theoretical model, no mechanisms intrinsic to the array were found to induce phase locking between rows of junctions. The results of these calculations and their impact on and relevance to the design of phase-locked Josephson oscillators are discussed.

Phase-locking of disordered two-dimensional Josephson junction arrays to microwave radiation

“…Unfortunately, it appears that some of the problems are still present in two-dimensional arrays. For instance Wiesenfeld, Benz, and Booi have shown that there is a transformation that completely decouples the 2D bare arrays in independent oscillators [1]. In spite of the simplifying assumptions used to perform the transformation (essentially to neglect disorder and inductive e®ects) a decoupling of 2D arrays in rows weakly interacting [9,10] or not interacting [11] has shown good e±cacy when compared with numerical simulations that include, to some extent, inductive e®ects.…”

“…The basic idea is that by studying the phase lock to an external radiation one can infer on the possibility of spontaneous phase lock among the junctions. The importance of spontaneous phase lock is twofold: on one hand for applications because the power emitted by a single junction is not enough for many practical uses [1]. On the other hand, from a theoretical point of view, the phenomenon of entrainment of disordered oscillators has led to many important discoveries in nonlinear dynamics [2].…”

Effect of cross-type bias in a two-dimensional array of short Josephson junctions

“…Analytic and numerical analysis have even hinted that in some range of the parameters disorder can enhance the coupling. 10,11 On the other hand, it has been demonstrated, 12 when neglecting inductance effects, that the in-phase solution of two-dimensional arrays suffers from some of the same problems as those found in phase locking of series arrays. The key point of this problem is that when each row of the twodimensional array is oscillating in phase ͑i.e., all the vertical junctions in a row are phase locked͒, the horizontal junctions are inactive and there is no coupling mechanism between rows.…”

Self-field effects in Josephson junction arrays

“…1 In fact, it is well known that the power available from a single junction is not enough for many practical applications, and therefore the achievement of coherent motion of arrays of junctions is an important issue for device applications. 2 Apart from the applications, two-dimensional arrays have been also investigated as an interesting nonlinear system both experimentally and theoretically. 3,4 In the study of such arrays two classes of models have been proposed to describe the dynamics: ͑1͒ Models that neglect the self-field effects.…”