Fig. 3: Experimental Floquet metal–insulator transition and Floquet Hofstadter butterfly in the AAH model.

a, Upon a single-site excitation of the Hermitian quasicrystal, the spatial spreading of the wavefunction is shown via the second moment M₂ of its position operator after a large propagation time of m = 200. The grey area marks the tolerance region of expected deviations owing to limited accuracy in the lattice parameters (Methods). One can see that a drastic spatial localization of the wavefunction sets in upon increasing the intersite coupling parameter beyond β_c = π/4 (top). This is exactly the metal–insulator phase transition, known from the static AAH model. b, At the symmetry point β = π/4, the Floquet Hofstadter–Harper model emerges. The evaluation of the quasienergies θ at m = 380 for 200 different phase gradients φ yields the Floquet Hofstadter butterfly (bottom) (Supplementary Section 3). The large propagation time allows for the high energy resolution. Compared with the original Hofstadter butterfly, our Floquet butterfly appears to be horizontally squeezed, due to the 2π periodicity in θ. The distribution of eigenvalues θ is obtained by applying the temporal Fourier transform (FT) to \({u}_{0}^{m}\). Here, \(\left|{u}_{0}^{m}\right|\) is retrieved from the intensity measurement. The phase information, which is only a ± sign here, is lost in the intensity measurement, and we therefore added this minor information to the experimental data based on equation (1). The Floquet butterfly without this sign information is shown in Supplementary Section 3.