Investigation of Band Structure Topology via Directly Probing a Dirac Point in an Optical Honeycomb Lattice

In a system with band structure, a point where two bands become degenerate marks a singularity in the Bloch wavefunction, where it cannot be uniquely determined. Studying such singularities provides a path toward understanding the geometry and topology of the Hilbert space spanned by the Bloch wavefunctions. Several experiments have observed effects of the Abelian Berry connection in a system with a Dirac point (a linear band-touching point), such as the accumulated Berry phase of π after enclosing a Dirac point in a single path-independent loop in quasimomentum. Instead, we study effects of the Berry connection by probing the Dirac point in a new way with a particular set of quasimomentum trajectories. To study this physics, we load a Bose-Einstein condensate into an optical honeycomb lattice with Dirac points, and use feedback control to dynamically translate the lattice in two-dimensions, allowing us to move the atoms along some quasimomentum trajectory. For straight trajectories from zero quasimomentum to the Dirac point, which then turn and exit at some angle, we observe band population transfer that is consistent with non-interacting band-theory. We find that this transfer depends on the exit angle and is independent of the speed (only up to a certain speed) that the trajectory is traversed, indicating that this behavior is the result of band structure topology and not dynamics.