Bloch dynamics in lattices with long-range hopping

Abstract

We study a discrete Schrödinger equation with arbitrary long range hopping terms under the influence of an external force. The impact of long range hoppings on the single particle Bloch dynamics in the lattice is investigated. A closed expression for the propagator is given, based on which we analyze the dynamics of initially Gaussian wave packets. Our findings capture the anharmonic oscillations recently observed in zigzag lattices and furthermore provide a detailed quantitative description of the crossover between center of mass Bloch oscillations for wide wave packets and left-right symmetric width oscillations for narrow single site excitations. The analytical results are shown to be in agreement with numerical simulations. A helix lattice setup for ultracold atoms is proposed where such hopping terms to far neighbors can be experimentally tuned to sizable values.

Nonlocal discrete continuity and invariant currents in locally symmetric effective Schrödinger arrays

“…Multidimensional divergence naturally prevents individual q S nm from being spatially constant along a selected dimension, just as is the case for the usual current. NLC invariance does apply if the sites m above represent next-nearest or (selected) remote neighbors along a single dimension, as is effectively the case, e. g., for zigzag chains with embedded scatterers [71] or discrete helical structures [72], respectively. Extended locally symmetric domains in such setups will then be distinguished by constant NLCs in arbitrary stationary states.…”

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

“…While in this work we have restricted ourselves to commensurate helix geometries, accounting only for a single inter-winding hopping term, it can be expected that the methods outlined here are relevant to a wider class of discrete nonlinear Schrödinger models with isolated longrange hopping terms. A particularly interesting direction for future studies is an extension to multi-strand helix lattices, which arguably are more accessible for implementations in ultracold-atom experiments than the single-strand helix [13,23]. Furthermore, given the emergence of the helicoidal discrete nonlinear Schrödinger equation as an envelope approximation of more complex nonlinear lattice systems, for instance of the extended Peyrard-Bishop model of DNA [40], a modulational-instability based analysis may also provide valuable information on the properties of localized solutions in such models.…”

“…For simplicity, we focus here on certain commensurate lattice geometries with N sites per winding in which only hopping to a single site on each of the neighboring windings is accounted for. Crucially, invariance of the helix lattice under discrete screw operations leads to an effective translational invariance of the model, in that the spatial difference between two sites only depends on their index difference [13], justifying the use of site-independent hopping parameters t 1 , t N in Eq. (1).…”

“…However, three-dimensional potential landscapes with deep minima at the desired helix lattice sites could be tailored for ultracold bosonic atoms, cf. [13,23] (see also [59] for a recent proposal of zigzag optical lattices). In this framework, a DNLS description arises in the mean-field treatment of the lowest-band tight-binding model, with Ψ j denoting the local condensate order parameter at site j [60,61].…”

“…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]. The recently proposed experimental implementation of helix-shaped optical traps for ultracold atoms [23] calls for a deeper understanding of the bosonic quantum many-body physics in helical geometries.…”