Fig. 2: Band structure spectroscopy of the 2D Chern insulator with a synthetic dimension.

a Typical data of \(\langle {\hat{\sigma }}^{x}\rangle \) and \(\langle {\hat{\sigma }}^{y}\rangle \) versus time t when choosing Q₈ as the target qubit for \(b=\frac{1}{3}\), Δ/2π = 12 MHz and \(\phi=\frac{2\pi }{3}\). b Squared FT magnitudes ∣A_j∣² of the response functions \({\chi }_{j}(t)\equiv \langle {\hat{\sigma }}_{j}^{x}(t)\rangle+i\langle {\hat{\sigma }}_{j}^{y}(t)\rangle \) for all fifteen qubits. Note that the edge states at the edge qubits: Q₁ and Q₁₅. c Summation of the squared FT magnitudes I_ϕ ≡ ∑_j∣A_j∣². d Band structure of the 2D Chern insulator for \(b=\frac{1}{3}\) and Δ = 12 MHz with 15 sites along the x-direction and periodic along the y direction. Red curves show the edge states. e, f Experimentally measured data for I_ϕ (e) for different values of ϕ varied from 0 to 2π are compared with numerically calculated data of I_ϕ (f) obtained by numerically simulating the dynamics of the 15-qubit system without decoherence.