Wave phenomena in Josephson junction ladders: breathers, flat bands and more

Abstract

Ladders of inductively coupled small Josephson junctions (JJs) support a number of nontrivial wave phenomena. Spatially localized excitations known as intrinsic localized modes or discrete breathers have no analog in the continuum limit. These excitations are whirling states which persist under a spatially homogeneous force due to the bias current. Their existence has been confirmed experimentally [1]. The stability analysis of these solutions is used to interpret the measured patterns in the I-V characteristics [2]. The dispersion law for the Josephson plasmons in the quasi-one-dimensional multiladder-like arrays that consists of the finite number of N rows (N>1) in Y direction and is infinite in X direction has 2N − 1 branches [3]. Among these branches there is an N-fold completely flat degenerate one that coincides with the Josephson plasma frequency. The remaining N − 1 branches have a standard Josephson plasmon dispersion law typical for parallel 1D JJ arrays. Application of the uniform dc bias on the top of each vertical column of junctions lifts the degeneracy and only one flat branch remains unchanged. The rest of the previously flat branches become weakly dispersive. The parameter range where the flatness of these branches is maximal has been discussed. The density of states of the Josephson plasma waves has a δ-function term due to the flat band and 3N−2 singularities [4]. The spatial distribution of the amplitudes of the plasmon wave is expressed through the orthogonal polynomials that are similar but not identical to the Chebyshev polynomials. Topological solitons known as fluxons or Josephson vortices exist in arrays of Josephson junctions. Comparison between fluxons in ladders and simple 1D JJ arrays is performed including the respective I-V characteristics.