Bloch sphere description of multi-level systems: Chern number and state tomography

The geometric interpretation of spin 1/2 systems on the Bloch sphere S² is well established in different areas across condensed matter physics and quantum information. These descriptions not only provide a visual aid for capturing various phenomena, but are also useful in bridging abstract concepts from mathematics to reveal hidden connections, such as novel topological relations, in experiments. Although such notions are known in mathematics for larger Hilbert spaces, beyond spin 1/2 they have been so far favored less for practical usage in condensed matter. By extending this geometric description to multi-band models, we characterize their dynamics on the higher dimensional Bloch sphere. We show that the topological properties such as the Berry curvature exhibit simple forms in terms of the coherence vector which is the object of interest on the Bloch sphere. We reveal an effective two-level description for the Chern number hidden among multi bands. We also present a tomography scheme to extract the coherence vector of the state in quantum simulation experiments via simple quenches, which gives access to these geometrical and topological properties by constructing the full wave function.