Observation of the Hopf Links and Hopf Fibration in a 2D topological Raman Lattice

Abstract

A dynamical Hopf insulator is experimentally synthesized with a quenched two-dimensional quantum anomalous Hall system on a square Raman lattice. The quench dynamics for the quasimomentum-time-dependent Bloch vectors defines a Hopf map from $(q_x,q_y,t)\in T^3$ to the Bloch sphere $S^2$. In this Hopf map, a dynamical Hopf number can be defined, and it exactly equals the Chern number of the post-quench Hamiltonian. We experimentally measure the Hopf link between the fibers for the North and South Poles on $S^2$, which are the trajectories in $(q_x,q_y,t)$ space with maximal spin polarization, to extract the topological Chern number of the post-quench Hamiltonian. We also observe the structure of Hopf fibration for the mutually nested Hopf tori. Our study sheds some new light on the interplay between topology (Hopf number) and geometry (fiber bundle) in quantum dynamics.