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 $\left[\right(2+1\left)\mathrm{D}\right]$, 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 $\pm 1$ 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 $(2+1)\mathrm{D}$ quantum dynamics.

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• Cold gases in optical lattices
• Spin-orbit coupling
• Synthetic gauge fields
• Topological phases of matter
• Bose-Einstein condensates
• Topological materials
• Ultracold gases
• Optical lattices & traps