Measuring the Chern-Simons invariant in quantum gases - Yi, Chang-Rui; Yu, Jinlong; Yuan, Huan; Chen, Xin; Guo, Jia-Yu; Zhang, Jinyi; Chen, Shuai; Pan, Jian-Wei (Physical review, 2025)
The Chern-Simons (CS) invariant is a fundamental topological invariant describing the topological invariance of three-dimensional (3D) space based on the Chern-Simons field theory. To date, direct measurement of the CS invariant in a physical system remains elusive. Here, the CS invariant is experimentally measured by quenching a 2D optical Raman lattice with 1/2 spin in ultracold atoms. With a recently developed Bloch state tomography, we measure the expectation values of three Pauli matrices in 2D quasimomentum space plus 1D time [(2+1)⁢D], and then respectively extract the Berry curvature and the corresponding Berry connection. By integrating the product of these two quantities, we obtain the CS invariants near ±1 and 0, which are consistent with theoretical predictions. We also observe transitions among these values, which indicates the change of the topology of the quantum state in (2+1)⁢D quantum dynamics.
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