Deshdeep Sahdev

Mode-locking, hysteresis and chaos in coupled Josephson junctions

Abstract We study dynamics of a triangular plaquette of underdamped Josephson junctions driven by DCs, at Zero temperature and in zero magnetic field. We plot the current-voltage characteristics numerically along three different directions in the relevant parameter space. In two of these, the I–V curve contains multiple hysteresis loops, while in the third, the system further displays the period-doubling route to chaos. We try to understand these I–V curves in terms of the phenomenon of mode-locking and of the existence of six special rays, on which the symmetry of the system forces mathematically constrained solutions. Finally, we interpret several aspects of the systems behaviour in terms of vortices.

The dynamical response of a three-junction network: I: phase variables

Abstract The phase diagram of a triangular network of overdamped Josephson Junctions driven by independent current drives is studied through simulations and analysis. We delineate regions of parameter space corresponding to different dynamical behaviour viz. steady, periodic and quasi-periodic. The steady-state boundary turns out to be the critical line on which the periodic tongues converge. We find the equation of this line and analyse the critical behaviour by making connection with the circle map. The correlation dimension of the set complementary to the periodic tongues on the critical line is also measured.

The dynamical response of a three-junction network II: vortices

Abstract The phase diagram of a triangular network of overdamped Josephson Junctions driven by independent current drives is studied in terms of vortices. There are no vortices in the fixed-point region, while in the p q Arnold tongue, vortices appear in sequences which repeat themselves every q vortices. Precisely q − p vortices in each sequence are injected by the non-uniform drive. We explicitly identify the vortex sequence at infinity and find that for large input currents, all reals between 1 and 1 2 can be given a unique binary representation in terms of vortices.

Fast algorithms for Josephson-junction arrays: Busbars and defects.

We critically review the fast algorithms for the numerical study of two–dimensional Josephson junction arrays and develop the analogy of such systems with electrostatics. We extend these procedures to arrays with bus–bars and defects in the form of missing bonds. The role of boundaries and of the guage choice in determing the Green’s function of the system is clarified. The extension of the Green’s function approach to other situations is also discussed. The dynamical properties of Josephson junction arrays (JJAs) are currently the foci of several experimental and theoretical investigations [1]. These arrays can now be routinely fabricated in several sizes and geometries, and the characteristics of their junctions can be varied at will over a wide range of values [2]. A large body of high–precision experimental data has consequently become available for JJAs in the presence of external magnetic fields [3–5]. On the theoretical front, several insights into the behaviour of JJAs have come from numerical studies of the underlying equations of motion as given by the resistively- and capacitively-shunted junction (RCSJ) model using input current drives and defects, both controlled [6] and random [7]. With the size of experimental arrays increasing continuously, and with the number of interesting effects best seen only in large arrays going up in equal measure, it has become imperative to find ever more efficient algorithms for implementing the corresponding simulations, inclusive of all the experimental conditions. An example of the latter for current–driven arrays is the presence of bus-bars, through which the external current can be conveniently injected or withdrawn. To understand the problem which these algorithms must address, we recall that in the RCSJ model, the total current, iij, (inclusive of external drives where applicable) flowing through the junction between sites i and j, is viewed as consisting of three ‘channels’ in parallel: superconductive, resistive ( or ohmic) and capacitive. The currents in each of these channels can be expressed in terms of the phase difference, θij = θi − θj, across the junction. This leads to the following equation for the evolution of the latter in time: C¯

VORTEX PINNING AND CRITICAL CURRENTS IN JOSEPHSON JUNCTION ARRAYS WITH DEFECTS

We develop an algorithm based on the Fast Cosine Transform to study two-dimensional arrays of Josephson junctions containing defects. We apply it to arrays as large as 128 × 256 and study vortex pinning, the transition from the superconducting to the resistive state, and various finite size effects. We find that the pinning potential for vortices is highly anisotropic in rectangular arrays due to boundaries and finite size effects. As a result we observe pinned vortices in arrays much smaller than expected so far. The energy of an array changes discontinuously at transitions from one vortex sector to another in the steady-state regime.

Vortex–Vortex Interaction Mediated Motion of Vortices in Josephson Junction Arrays

We numerically study vortex motion in arrays of over- and under-damped Josephson junctions. We show quantitatively that vortex–vortex interactions (VVI) in a vortex-street are large enough to overcome the pinning potential and cause vortices to move even in a region with no external current. The experiments of van der Zant et al.1 are reinterpreted in terms of VVI without invoking ballistic motion of vortices. The discrepancy with simulations which have all failed to observe any ballistic motion is thus removed. Our results also indicate that the vortex mass is negligible for the McCumber–Stewart parameter, βc < 100.