Dynamics of Josephson Junction Arrays.

The dynamics of Josephson junction arrays is a topic that lies at the intersection of the fields of nonlinear dynamics and Josephson junction technology. The series arrays considered here consist of several rapidly oscillating Josephson junctions where each junction is coupled equally to every other junction. The purpose of this study is to understand phaselocking and other cooperative dynamics of this system. Previously, little was known about high dimensional nonlinear systems of this sort. Numerical simulations are used to study the dynamics of these arrays. Three distinct types of periodic solutions to the array equations were observed as well as period doubled and chaotic solutions. One of the periodic solutions is the symmetric, in-phase solution where all of the junctions oscillate identically. The other two periodic solutions are symmetry-broken solutions where all of the junctions do not oscillate identically. The symmetry-broken solutions are highly degenerate. As many as (N - 1)! stable solutions can coexist for an array of N junctions. Understanding the stability of these several solutions and the transitions among them is vital to the design of useful devices. From the technological point of view the most useful dynamical state of the junction arrays is the in -phase state where all of the junctions oscillate identically. A detailed analysis of the stability of the in-phase state is given and the fluctuations about the in-phase state are described. Using this analysis a proposal is made for the design of a generator of millimeter wave radiation that maximizes the stability of the in-phase state. The other technological application that is discussed is parametric amplification. The relation between the instabilities of this system and the process of parametric amplification is described and a proposal is made for the design of a high gain parametric amplifier that exploits a previously undocumented instability of this system.