Fig. 2: Energy eigenspectrum, edge modes, and corner modes of the synthetic dimension quadrupole HOTI.
From: Higher-order topological insulators in synthetic dimensions
a–c Schematics of selective excitation of corner, edge, and bulk modes in the photonic molecule array using external waveguides. d Intensity \(\left| a \right|^2\) in the excited rings under various input frequency detunings Δω for the excitations indicated in a–c. Peaks appear at input frequency detunings that correspond to the positions of the corner, edge, and bulk modes in e. Solid lines represent tight-binding model solutions under the rotating wave approximation [RWA, Eq. (2)], with γ/λ = 0.1. Dots represent the solutions of the full dynamical coupled-mode equations using the modulation in Eq. (1), with A1/K = 0.05 and A0/A1 = γ/λ = 0.1 e Energy eigenspectrum for a large finite lattice of the quadrupole HOTI. For \(\left| {\gamma /\lambda } \right| < 1\), the system exhibits topologically protected corner modes pinned to zero energy. For \(\left| {\gamma /\lambda } \right| > 1\), the corner modes cease to exist. f Same as in d but in the topologically trivial regime γ/λ = 1.1. No peaks are observed in the bandgap, as the corner and edge modes cease to exist. The overall amplitude in the excited rings is lower than that in c because the excitation spreads into the bulk. g–i Cavity field intensity when exciting the finite lattice at the corner g, edge h and bulk i for γ/λ = 0.1. The RWA results agree well with the solution of the full dynamical equations. No such corner or edge localized modes were observed in the trivial phase under the same excitations for \(\left| {\gamma /\lambda } \right| > 1\). The color scale in the bottom two rows indicates the steady-state field amplitude distribution in the lattice. Blue dashed circles denote the lattice site excited in each case. j Lattice field distributions for corner excitation in the topological phase with γ/λ = 0.1 obtained using the full dynamical equations for increasing values of A1/K = 0.2, 1, and 2, all beyond the validity of the RWA [Eq. (2)]. The field distribution significantly deviates from that for the corner mode excitation based on the RWA (in g), but corner localization is still observed for moderate modulation strengths A1/K < 1. k Simulations highlighting the difference in the field distributions obtained using the full solution and the RWA upon exciting a corner site in the trivial regime [A0/A1 = γ/λ = 1.1] under ultrastrong modulation [A1/K = 1]
