Nested-sphere description of the N-level Chern number and the generalized Bloch hypersphere

TL;DR

This paper introduces a geometric framework using nested spheres to interpret the topological properties of N-level quantum systems, extending the Bloch sphere concept beyond two levels and enabling experimental measurement of topological invariants.

Contribution

It develops a generalized Bloch hypersphere description for N-level systems, revealing a nested sphere structure that characterizes the Chern number and facilitates experimental probing.

Findings

Nested sphere structure encodes the N-level Chern number

External two-sphere measurement relates to topological properties

Proposed tomography scheme for three-level systems

Abstract

The geometric interpretation of (pseudo)spin 1/2 systems on the Bloch sphere has been appreciated across different areas ranging from condensed matter to quantum information and high energy physics. Although similar notions for larger Hilbert spaces are established in mathematics, they have been so far less explored beyond the two-level case for practical usage in condensed matter settings, or have involved restrictions to sub manifolds within the full Hilbert space. We here employ a coherence vector description to theoretically characterize a general N-level system on the higher dimensional generalized Bloch (hyper)sphere by respecting the structure of the underlying SU(N) algebra and construct physically intuitive geometric pictures for topological concepts. Focusing on two spatial dimensions, we reveal a geometric interpretation for the Chern number in larger Hilbert spaces in terms of a nested structure comprising N-1 two-spheres. We demonstrate that for the N-level case, there is an exterior two-sphere that provides a useful characterization of the system, notably by playing a primary role in determining the Chern number. The external sphere can be directly measured in ultracold atoms via well-established band mapping techniques, thereby imparting knowledge of the topological nature of state. We also investigate how the time evolution of the coherence vector defined on the generalized Bloch hypersphere can be utilized to extract the full state vector in experiments, allowing us to develop a tomography scheme involving quenches for three-level systems. Our geometric description opens up a new avenue for the interpretation of the topological classification and the dynamical illustration of multi-level systems, which in turn is anticipated to help in the design of new experimental probes.