Dirac string and optical excitation selection rules originating from a full inter-band Berry connection measurement in the optical honeycomb lattice

The inter-band Berry connection is closely related to the transition matrix element allowing for (optical) excitations in solids. Its nature as a vector field allows it to encode polarization selection rules, in addition to the transition strength. Lattice structures like the optical honeycomb exhibit band crossing points, which can cause the necessity for the introduction of a Dirac string in the inter-band Berry connection. We measure the full real vector field (up to a gauge choice) for multiple ground to excited bands using a Degenerate Fermi Gas of ⁴⁰K in a 8.95E_r (E_r = h*4.41kHz is the recoil energy) deep optical honeycomb lattice. We induce transitions by shaking the lattice in different directions linearly and circularly. We find polarization transparency lines in quasi-momentum space that are anchored at high-symmetry points. In addition, our measurement of the inter-band Berry connection between the ground and second excited band requires the introduction of a Dirac string. This work paves the way towards the study of multi-gap topology, as the real inter-band Berry connection of neighboring bands, sometimes referred to as Euler connection, can be used to measure the Euler class.