Josephson Lattices of the Optimal Size

TL;DR

The paper addresses the stability of bound states of magnetic flux in a one-dimensional Josephson junction array with a periodic lattice of resistive inhomogeneities. The governing equation is the stationary perturbed sine-Gordon equation $-\phi_{xx}+j_D(x)\sin\phi+\gamma=0$ on a finite interval, and the inhomogeneity profile is modeled by trapezoids with width $\mu$, top width $\sigma=m\mu$, separated by $\Delta$. Stability is assessed by solving the Sturm-Liouville problem $-\psi_{xx}+q(x)\psi=\lambda\psi$ with $q(x)=j_D(x)\cos\phi(x)$ and Neumann-type boundaries, via a continuous Newton method with spline-collocation. A key result is the identification of an optimal lattice geometry that maximizes the interval of current stability $\Delta\gamma$, found by analyzing the dependence of the minimal eigenvalue $\lambda_{min}$ on the lattice width $\mu$ and spacing $\Delta$, revealing an approximately linear relation $\mu_s(\Delta) \approx a\Delta+b$ with $a\approx0.254$, $b\approx-8.5\times10^{-3}$ and $\Delta_{opt}\approx5.2$ for $N_I=2$. These findings provide design guidelines for robust, periodically pinned flux states in Josephson lattices and document the presence of both periodic and aperiodic bound states and their bifurcations.

Abstract

The stability of the bound states of the magnetic flux in a Josephson resistive lattices is investigated numerically. It is shown that for a simple relationship between the geometrical parameters of the lattice the range of bias current is the widest.