Direct Characterization of Dirac Points in an Optical Honeycomb Lattice

We study properties of singularities in the band structure of a honeycomb lattice by probing the singular points directly with a BEC in an optical lattice. In a periodic system, if the band structure becomes degenerate at some quasimomentum, that point is called singular if there are no choice of gauge such that the Bloch wave function is continuous there. These singularities reveal important topological properties of the band structure, quantified by the Berry connection. In the past, the Dirac points of a honeycomb lattice have been studied by, for example, accelerating a BEC in the momentum space and enclosing the singularity in the resulting path. Here we study the singularities by directly accelerating a BEC to the singular point and making a turn with varying angle in the momentum space. Our initial results show that the population transfer after such a trajectory is close to the value predicted by a noninteracting band theory. Moreover, for a specific final quasimomentum, the population transfer is significantly less dependent on the rate we accelerate the atoms when we go past the singularity, compared to a straight path that avoids the singularity. This indicates that the transfer observed at Dirac point is mainly determined by the band topology instead of dynamics.