Fig. 4: Experimental triple phase transition in a non-Hermitian Floquet quasicrystal.

a, Three simultaneous phase transitions are shown from top to bottom. The two regimes of the triple phase transition are separated from left to right by the vertical dashed line. Top: for β < β_c (left), the quasicrystal is expected to be in a topologically non-trivial phase, owing to the formation of point-energy gaps with non-zero winding w = 1. The quasienergy spectra are obtained from numerical diagonalization of the Floquet propagator with periodic boundary conditions. For β > β_c (right), the topological phase changes, as the spectrum becomes real and the winding changes. Centre: for β < β_c (left), the quasicrystal is in the broken PT phase, which is marked by the exponential growth λ of the overall energy \({{\sum }_{n}|{u}_{n}^{m}|}^{2}+{|{v}_{n}^{m}|}^{2}\propto {{\rm{e}}}^{{\rm{\lambda }}m}\) in time. For β > β_c (right), the system changes to the unbroken PT phase, where the spectrum becomes real, and the overall energy becomes on-average constant. Bottom: for β < β_c (left), the quasicrystal is in the delocalized phase, which is marked by a monotonic increase of the second moment that indicates strong spatial spreading of the wavefunction. For β > β_c (right), all eigenstates become exponentially localized, which is marked by the extremely low and bounded second moment. All experiments are based on single-site excitations. The experimental data agree well with the predicted transition point at β_c = 0.275π. The grey areas mark the tolerance regions of expected deviations owing to limited accuracy in the lattice parameters (Methods). b, Although a direct measurement of the winding number is not possible with the experimental setup, we observe light localization at a topological interface (top, β₁ = 0.70β_c and β₂ = 1.03β_c compared with a trivial interface (bottom, β₁ = 0.70β_c and β₂ = 0.89β_c), where light does not localize at the interface. The localization at the interface vanishes, as soon as the right medium exceeds critical coupling β_c, such that both sides would have the same topological winding w = 1.