Non-Abelian non-adiabatic geometric manipulation of spinor BEC

Adiabatic non-Abelian geometric operations play a crucial role in quantum systems due to their inherent robustness. However, their implementation requires long evolution times, posing significant challenges for maintaining coherence in practical quantum systems. In this work, we experimentally realize non-adiabatic non-Abelian geometric manipulations in a spinor Bose-Einstein condensate (BEC) system. We demonstrate that two distinct quantum operations, originating from different paths in Grassmann space, yield non-equivalent transformations when applied in different orders, directly showcasing their non-Abelian nature. Notably, these geometric operations are non-adiabatic and exhibit high-speed dynamics. Our results pave the way for robust geometric control of spinor BEC, with promising applications in quantum computation and quantum simulation.