Compact stationary fluxons in the Josephson junction ladder

TL;DR

This work demonstrates the existence of exact stationary compact fluxons in a dc-biased Josephson junction ladder, a phenomenon absent in standard parallel arrays governed by the discrete sine-Gordon equation. By formulating a two-row JJL with vertical and horizontal junctions and deriving a DSG-like, dimensionless equation set, the authors identify two classes of compact fluxons (antisymmetric and asymmetric) and derive explicit energy expressions $E_{ANS}[Q]$ and $E_{A-S}[Q]$, revealing zero-energy states for certain charges when $\gamma=0$. A key result is the existence of a critical anisotropy $\eta_c(\beta_L)$ below which compact fluxons are energetically favorable and stable, while above this threshold non-compact fluxons persist; a dc bias and the flat-band structure underpin these stationary states. Numerical simulations corroborate the analytical energy relations, map the transition between compact and non-compact states, and show how decreasing $\eta$ localizes fluxons further. The study also shows that compact fluxons interact with plane-wave plasmons, scattering while maintaining localization, and that external magnetic fields destroy compactness, highlighting the delicate balance between lattice geometry, bias, and field in these topological excitations. Overall, the work reveals a new, parameter-tunable class of stationary topological excitations in JJLs with potential implications for controlled localization and plasmon–fluxon interactions in superconducting networks.

Abstract

Stationary compact fluxon profiles are shown to be exact solutions of the inductively coupled and dc-biased Josephson junction ladder. Such states do not exist in the parallel Josephson junction array which is described by the standard discrete sine-Gordon equation. It is shown that there are compact fluxon and multi-fluxon states which either satisfy the top-bottom antisymmetry or are asymmetric. The anti-symmetric states have zero energy if their topological charge is even and the asymmetric states always have zero energy. Depending on the anisotropy constant the compact fluxons can either coexist with the non-compact states or only compact states are possible. External magnetic field prevents compact state existence.

Figures (9)

• Figure 1: Schematic view of the dc-biased 2-row ladder with open ends. Crosses correspond to the Josephson junctions.
• Figure 2: Fluxon profile compactification for the vertical (a) and bottom horizontal (b) junctions as a function of time for the discreteness $\beta_L=1.5$, bias $\gamma=0$ and anisotropy $\eta=0.5$. The ladder size is $N=40$.
• Figure 3: Time evolution of the $(n_0-1)$ vertical junction for the case described in Fig. \ref{['fig2']}.
• Figure 4: Fluxon profile compactification for the vertical (a), horizontal bottom (b) and top (c) junctions as a function of time for the same parameters as in Fig. \ref{['fig2']}.
• Figure 5: (a) Energy of the non-compact stationary fluxons as a function of the anisotropy $\eta$ for $\beta_L=0.25$ (line 1, red), $\beta_L=1$ (line 2, black) and $\beta_L=3$ (line 3, blue). The straight line ($\blacklozenge$) corresponds to the energy of the compact fluxon, $E=4\eta$. (b) Maximal value of the horizontal bottom phase, $\phi^{(h)}_{n_0,1}$ as a function of anisotropy for $\beta_L=1$. The dashed red line corresponds to the value $\phi^{(h)}_{n_0,1}=\pi$. The insets demonstrate the vertical $\phi^{(v)}_{n}$ and bottom horizontal $\phi^{(h)}_{n,1}$ distributions for $\eta=5$ ($\blacklozenge$), $\eta=2$ ($\circ$) and $\eta=0.8$ ($\oplus$). The markers on the curve correspond to the above metioned values of $\eta$.