Synthesizing arbitrary dispersion relations in a modulated tilted optical lattice

Dispersion relations are fundamental characteristics of the dynamics of quantum and wave systems. In this work we introduce a simple technique to generate arbitrary dispersion relations in a modulated tilted lattice. The technique is illustrated by important examples: the Dirac, Bogoliubov and Landau dispersion relations (the latter exhibiting the roton and the maxon). We show that adding a slow chirp to the lattice modulation allows one to reconstruct the dispersion relation from dynamical quantities. Finally, we generalize the technique to higher dimensions, and generate graphene-like Dirac points and flat bands in two dimensions.