Fig. 3: Experimental obtained band structures of the Creutz ladder and direct observation of the Aharonov–Bohm cage effect.

a Illustration of the Creutz ladder. b, c The heat maps display two general band structures of the Creutz ladder given different ϕ₁ and ϕ^C, where J^V/2J^H = 1.1, J^C/J^H = 0.52 and J^H = 0.06Ω. d The flat band structure measured at J^V = 0, J^H = J^C = 0.028Ω and \(-{\phi }_{A}^{H}={\phi }_{B}^{H}=\pi /2\). The absence of J^V, equal J^H and J^C, together with the π flux also satisfy the Aharonov–Bohm cage effect condition. The left panels in (b–d) are the experimental results and the right panels are the numerical calculations. The population information on the two spins (resonators) are encoded in the normalized intensities. e Illustration of the Aharonov–Bohm cage effect where the phase collected in a round trip in the pink area is π. When a probe light is input from resonator A with frequency near the zero index mode, the distribution is caged within the dashed box (0, ±1 in lattice A and  ±1 in lattice B). f, g The blue points are the experimental readout of the mode distribution from the drop ports of resonator A and B measured by a Fabry–Perot (F-P) cavity. The pink dashed lines represent the normalized numerical calculations, where Δω is fitted to be approximate 4J^H. The distribution other than those inside the dashed box is clearly suppressed. h Illustration of the not caged situation where the phase collected in the pink area is zero. The light normally spreads out along the frequency direction. i, j The blue points are experimental results of the not caged situation, where more modes survives compared to the caged ones. The pink circles and dashed lines represent the normalized numerical calculations, where Δω is fitted to be approximate 2J^H. The distribution results are plotted in the logarithmic coordinates and the linewidth of the F-P cavity is taken into account.