  Ultracold atoms in optical lattices are pristine model systems with a
tunability and flexibility that goes beyond solid-state analogies, e.g.,
dynamical lattice-geometry changes allow tuning a graphene lattice into a
boron-nitride lattice. However, a fast modulation of the lattice geometry
remains intrinsically difficult. Here we introduce a multi-frequency lattice
for fast and flexible lattice-geometry control and demonstrate it for a
three-beam lattice, realizing the full dynamical tunability between honeycomb
lattice, boron-nitride lattice and triangular lattice. At the same time, the
scheme ensures intrinsically high stability of the lattice geometry. We
introduce the concept of a geometry phase as the parameter that fully controls
the geometry and observe its signature as a staggered flux in a momentum space
lattice. Tuning the geometry phase allows to dynamically control the sublattice
offset in the boron-nitride lattice. We use a fast sweep of the offset to
transfer atoms into higher Bloch bands, and perform a new type of Bloch band
spectroscopy by modulating the sublattice offset. Finally, we generalize the
geometry phase concept and the multi-frequency lattice to three-dimensional
optical lattices and quasi-periodic potentials. This scheme will allow further
applications such as novel Floquet and quench protocols to create and probe,
e.g., topological properties.
