Characterizing Topological Nodal Structures and Their Evolution With a Bose-Einstein Condensate

Characterizing topological nodal structures and how they evolve or even change their topology when parameters are continuously varied is important to the studies of topological states of matter but is challenging in experiments. Here, we observe the emergence of topological nodal rings and lines in the parameter space of the light (rf/microwave) fields that cyclically couple four atomic hyperfine spin states in a Bose-Einstein condensate. When the parameters are changed, the evolution of these nodal structures is probed by quench dynamics and Fourier spectroscopy. Our experimental results are consistent with the theory, which predicts that the nodal ring can change size and the two nodal lines can move and reconnect, but they can never be removed. Such evolution can be understood from the projection of a nodal hyperboloid into lower dimensions. This higher-dimensional perspective further reveals not only how the topology of the nodal rings and lines may be protected by the geometry of the nodal surface, but also how their topology may change, such as two nodal lines evolving into a ring. We also propose different light coupling schemes to create other nodal surfaces, whose projection reveals different evolution, such as creation and annihilation, of the lower-dimensional nodal structures. Our work provides experimental tools and further insights to study topological nodal structures and their evolution, including topological changes, in synthetic quantum matter.