Nonlinear dynamics of Aharonov-Bohm cages

Abstract

The interplay of $\ensuremath{\pi}$-flux and lattice geometry can yield full localization of quantum dynamics in lattice systems, a striking interference phenomenon known as Aharonov-Bohm caging. At the single-particle level, this full-localization effect is attributed to the collapse of Bloch bands into a set of perfectly flat (dispersionless) bands. While interparticle interactions generally break the cages, not much is known regarding the fate of Aharonov-Bohm caging in the presence of classical nonlinearities, as captured by a discrete nonlinear Schr\"odinger equation. This scenario is relevant to recent experimental realizations of photonic Aharonov-Bohm cages, using classical light propagating in arrays of coupled waveguides. In this article, we demonstrate that caging always occurs in this nonlinear setting, as long as the nonlinearities remain local. As a central result, we identify special caged solutions that are accompanied by a breathing dynamics of the field intensity that we describe in terms of an effective two-mode model reminiscent of a bosonic Josephson junction. Also, motivated by a formal similarity with the Gross-Pitaevskii equation describing interacting bosons, we explore the quantum regime of Aharonov-Bohm caging using small ensembles of interacting particles, and reveal quasicaged collapse-revival dynamics. The results stemming from this work open an interesting route towards the characterization of nonlinear dynamics in interacting flat-band systems.