Sub- and supercritical defect scattering in Schrödinger chains with higher-order hopping

Abstract

We theoretically analyze a discrete Schrödinger 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.

Modulational instability and localized breather modes in the discrete nonlinear Schrödinger equation with helicoidal hopping

“…Eqs. (12)- (15) are also in qualitative (and approximate quantitative) agreement with the numerical data and, in particular, correctly capture the trend that for N = 5 the critical norms P cr are substantially larger than for N = 2, while the frequencies µ cr shift closer to the linear limit, such that the P(|µ|) curves rise towards their maximum more steeply for N = 5.…”

“…In this appendix, we give the details of the derivation of Eqs. ( 12)- (15), providing approximate relations between |ψ 0 | 2 , µ and P for the near-linear breathers as encountered in the attractive model in Sec. 4, localized in the vicinity of k = 0 in quasimomentum space.…”

“…If the couplings originate from anisotropic (such as dipolar) interaction, the long-range terms can also be tuned via the orientations of the local oscillators [8]. A specific proposal for enhancing the second-neighbor hopping by arranging the lattice sites in a zigzag structure [9] stimulated numerous recent theoretical and experimental investigations in this direction, covering free expansion [10] and Bloch oscillation [11][12][13] dynamics, disorder-induced localization [14], defect scattering [15], nonlinear localized excitations [9,[16][17][18][19] and self-trapping [20] in such lattices. Generalizing this idea, a three-dimensional layout of lattice sites along a helix curve will lead to small inter-site distances (and thus large hopping probabilities) not only to the neighboring sites along the curve, but also to certain sites on the adjacent windings of the helix, admitting a geometry-induced enhancement of selected N -th neighbor hopping terms [13], see also [21,22].…”

“…Markers show the numerical results, solid lines for µcr, Pcr are obtained from Eqs. (14),(15), dashed lines from Eqs. (12),(13).…”

Engineering electronic states of periodic and quasiperiodic chains by buckling

“…Thus the essence of "buckling" is just to bring in the flavor of long range hopping in the system. Interestingly, similar long range hopping is considered recently in a Schrödinger chain, and has been shown to lead to a new extremum at the center of the density of states accompanied by a van Hove singularity [25].…”