Sub- and supercritical defect scattering in Schr\"odinger chains with higher-order hopping

TL;DR

This paper investigates how second-neighbor hopping in a Schrödinger chain affects band structure and scattering, revealing subcritical and supercritical regimes with unique wave phenomena and resonances.

Contribution

It introduces a detailed analysis of defect scattering in chains with higher-order hopping, highlighting new band features and resonance behaviors not seen in nearest-neighbor models.

Findings

Emergence of a new extremum inside the band due to second-neighbor hopping

Identification of Fano-Feshbach resonances at subcritical energies

Wave packet fragmentation and complex dynamics at supercritical energies

Abstract

We theoretically analyze a discrete Schr\"odinger chain with hopping to the first and second neighbors, as can be realized with zigzag arrangements of optical waveguides or lattice sites for cold atoms. Already at moderate values, the second-neighbor hopping has a strong impact on the band structure, leading to the emergence of a new extremum located inside the band, accompanied by a van Hove singularity in the density of states. The energy band is then divided into a subcritical regime with the usual unique correspondence between wave number and energy of the travelling waves, and a supercritical regime, in which waves of different wave number are degenerate in energy. We study the consequences of these features in a scattering setup, introducing a defect that locally breaks the translational invariance. The notion of a local probability current is generalized beyond the nearest-neighbor approximation and bound states with energies outside the band are discussed. At subcritical energies inside the band, an evanescent mode coexists with the travelling plane wave, giving rise to resonance phenomena in scattering. At weak coupling to the defect, we identify a prototypical Fano-Feshbach resonance of tunable shape and provide analytical expressions for its profile parameters. At supercritical energies, we observe coupling of the degenerate travelling waves, leading to an intricate wave packet fragmentation dynamics. The corresponding branching ratios are analyzed.