Josephson Junctions with Minimal Length

TL;DR

This work addresses the minimal spatial extent required to sustain stable bound states (fluxons) in one-dimensional Josephson junctions, including inhomogeneous cases. It recasts the problem as a closed nonlinear eigenvalue system combining the Josephson phase equation and its associated Sturm-Liouville stability operator, and computes bifurcation curves through a generalized Continuous Analog of Newton's Method to locate points where $\lambda_{\min}=0$. A key finding is the quantitative minimal length $\Delta_{\min}$ for a single fluxon (e.g., $\Delta_{\min}\approx 4.24$ for $h_B=\gamma=0$, reduced to $\approx 2.7$ for $h_B=1$) and the existence of an optimal inhomogeneity width $\mu$ near $1.1$ that minimizes $\Delta_{\min}$; boundary fields stabilize fluxons while currents destabilize them. The results offer design guidance for device sizing and reveal how inhomogeneity and boundary conditions shape fluxon stability, including shifts in optimal inhomogeneity location and the emergence of multi-soliton states under strong boundary fields.

Abstract

The minimal length of ``one-dimensional'' Josephson junctions, in which the specific bound states of the magnetic flux retain their stability is discussed numerically. Thereby, we consider as ``long'' every Josephson junction, in which there exists at least one nontrivial stable distribution of the magnetic flux for fixed values of all the physical and the geometrical parameters. Our results can be applied for optimization of the sizes of devices containing Josephson junctions for different operating conditions.