Fig. 2: Numerical simulation of Thouless pumping in tilted moiré lattices.

Pythagorean angle \(\theta=\arctan (4/3)\) (m = 2, n = 1) and tilt angles α = 0.002 (a, b, e) and α = 0.015 (c, d, f, g). a, c Displacements of the light beam COM, ∣r_c∣, with distance z. b, d Trajectories of the COM r_c = (x_c, y_c) in the transverse plane. The asterisks correspond to the points z = Z, i.e., to one longitudinal lattice period. Lines of different color correspond to different amplitudes E₀ of the moiré lattice. Dashed diagonal lines show the evolution of R_av(z) in (b) and (d), and of its modulus in (a) and (c) upon propagation. The results for classical motion in the potential U are shown by the dark-blue solid lines. The arrows along dotted lines in (b) and (d) show the direction of tilting of one sublattice with respect to another one. e Snapshots of beam propagation for α = 0.002 and E₀ = 10.0 (zoom of the initial intensity distribution is shown within white square). f, g Snapshots of beam propagation for α = 0.015 and E₀ = 7.5 and E₀ = 9.0, respectively. h Evolution of the mean radius of the beam propagating inside tilted moiré lattice (red line), a tilted lattice obtained without twisting of two sublattices (green line) with p = 0.3, and in a single reference lattice (blue line) with depth p = 1.3 for α = 0.015. In all cases, a = π, p = 0.3.