Observation of topological braiding and dynamical criticality in time reflection and refraction

Time reflection and refraction are among the most canonical phenomena emerging from the burgeoning field of time-varying materials. However, their connection with topology remains largely unexplored. Here, we experimentally report that temporal scattering processes exhibit two distinct topological characteristics: eigenstate topological braiding and dynamical topological phase transition. By developing approaches to implement Schrödinger dynamics and engineer temporal boundaries, various temporal scattering processes are implemented in circuit metamaterials. Through measurements of scattering coefficients, we observe diverse topological braiding and linking structures, whose linking numbers are governed by the difference in topological winding numbers before and after the temporal boundary. Furthermore, we reveal that these processes exhibit dynamical critical behavior and host dynamical topological phase transitions, evidenced by quantized jumps at critical time points in the measured dynamical topological invariants. Our study demonstrates fundamental links between temporal scattering and braiding and dynamical topology, opening avenues for exploring the topological aspects of time-varying and nonequilibrium phases of matter. Emulating Schrodinger-equation-governed temporal dynamics in metamaterials is of strong interest. Here, the authors demonstrate that temporal scattering processes exhibit eigenstate topological braiding and dynamical topological phase transitions in circuit metamaterials.