A sublattice polarimeter to probe the quantum geometry of synthetic photonic lattices

Measuring the electronic structure of crystalline materials is key in advancing our understanding of their physical properties. While techniques such as angle-resolved photoemission spectroscopy have provided unprecedented insights into electronic orbitals of natural crystals, an equivalent tool to characterize photonic lattices is still lacking. Here, we introduce a sublattice polarimeter that enables full reconstruction of the Bloch wavefunctions in bipartite photonic lattices. Using a spatial light modulator to apply controlled phase shifts and attenuations to each sublattice, we extract the amplitude and phase of the crystal orbitals across the Brillouin zone, including close to band degeneracies. This allows experimentally accessing the Bloch Hamiltonian, thereby determining the quantum geometric tensor and the band Chern numbers. Beyond bipartite lattices, the method can be generalized to address multiple internal degrees of freedom, opening routes to experimentally probe complex topological structures such as trigonal warping, Chern insulators and Euler curvature in multiband settings. The authors introduce a sublattice polarimeter to measure the Bloch wavefunctions in bipartite photonic lattices. Using their method, they experimentally access the quantum metric, Berry curvature and band Chern numbers.