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

a Typical data of $⟨{\widehat{\sigma }}^x⟩$ and $⟨{\widehat{\sigma }}^y⟩$ versus time t when choosing Q₈ as the target qubit for $b=\frac{1}{3}$, Δ/2π = 12 MHz and $\varphi =\frac{2\pi }{3}$. b Squared FT magnitudes ∣A_j∣² of the response functions ${\chi }_j\left(t\right)\equiv ⟨{\widehat{\sigma }}_j^x\left(t\right)⟩+i⟨{\widehat{\sigma }}_j^y\left(t\right)⟩$ 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.