Fig. 4. Experimental demonstration of the nonlinear control of a higher-order topological insulator.

a 3D view of a typical linear corner state experimentally observed in a nontrivial lattice. b, c Nonlinear self-focusing leads to b coupling into the edges (non-zero intensity along the edge sites compared with the linear case) when the nonlinearity is low and c a highly localized corner soliton when the nonlinearity is high. d Plot of the calculated nonlinear polarization as a function of the nonlinear control parameter γ_k as well as the dimerization parameter c. Characteristic jump in the bulk polarization, testifying that the topological phase transition extends beyond the linear regime (γ_k = 0) because of the inherited topology in the nonlinear regime. e, f Experimental results of the nonlinear control with a low and high self-defocusing nonlinearity. Under a high defocusing nonlinearity, the energy spreads dramatically to both the edge and the bulk (f). For the focusing (defocusing) case, the bias field is E₀ = 160 kV m⁻¹ (E₀ = −80 kV m⁻¹), and the average power of the probe beam is about 15 nW (70 nW) for the low (high) nonlinearity. See Supplementary Material for the corresponding numerical simulation results