Engineering the Bogoliubov Modes through Geometry and Interaction: From Collective Edge Modes to Flat-band Excitations

We propose a procedure to engineer solid-state lattice models with superlattices of interaction-coupled Bose-Einstein condensates. We show that the dynamical equation for the excitations of Bose-Einstein condensates at zero temperature can be expressed in an eigenvalue form that resembles the time-independent Schr{\"o}dinger equation. The eigenvalues and eigenvectors of this equation correspond to the dispersions of the collective modes and the amplitudes of the density oscillations. This alikeness opens the way for the simulation of different tight-binding models with arrays of condensates. We demonstrate, in particular, how we can model a one-dimensional Su-Schrieffer-Heeger lattice supporting topological edge modes and a two-dimensional Lieb lattice with flat-band excitations with superlattices of Bose-Einstein condensates.