Skin mode tunability and self-healing effect in photonic Floquet lattices

Abstract

Non-Hermitian systems host exotic phenomena absent in their Hermitian counterparts, including the recently predicted self-healing effect (SHE) of non-Hermitian skin modes. To date, the SHE of skin modes in non-Hermitian systems has not been observed experimentally. Here we propose a feasible scheme to realize SHE in photonic Floquet lattices by exploiting skin mode tunability (SMT), a mechanism in which the spectrum of skin modes localized at one boundary can be tuned via a potential applied at the opposite boundary. Such tunability arises from the non-Hermitian biorthogonality of the eigenstates. We demonstrate that a certain skin mode is exceptionally sensitive to remote-boundary potentials in an array of $100$ coupled helical waveguides, allowing broad-range spectral control and the generation of SHE with experimentally accessible parameters. Our results establish a general framework for engineering skin modes via local perturbations, thereby expanding the toolbox for non-Hermitian wave control.

Figures (5)

• Figure 1: (a) Tunability of a skin mode localized at the left boundary by a local perturbation acting on the right end. (b) Schematic of SHE, which demonstrates the wave function recovers to its original form after encountering a disturbance. (c) Mechanism of the SHE based on the SMT shown in (a), where the SHS is spectrally isolated with the largest imaginary eigenenergy, $\Im \epsilon_{j_0}$.
• Figure 2: (a) Floquet lattice for realizing the SHE in a zigzag helical waveguide array. Schematic representation of (b) the effective Hamiltonian under PBC, $\mathcal{H}_\text{F}^{\text{PBC}}$, (c) the effective Hamiltonian under OBC, $\mathcal{H}_\text{F}^{\text{OBC}}$, and (d) the OBC Hamiltonian with modified boundary term $\delta V_{\rm eff}$. Long-range interactions beyond next-nearest neighbors are not shown for clarity in (b)-(d).
• Figure 3: Amplitude (a) and phase (b) of the hopping term $t_{\pm 1,\pm 2}$ in the helical waveguides as a function of $R$. We choose $R_0=10.45 ~\mu \mathrm{m}$, where the first- and second-order coupling strengths are comparable, with $t_1/\kappa=0.128 + 0.016i$, $t_{-1}/\kappa=0.104 +0.068i$, $t_2/\kappa=-0.098i$, $t_{-2}/\kappa=0.074i$.
• Figure 4: (a) Energy spectrum of $\mathcal{H}_\text{F}^{\text{PBC}}$, which forms a loop and exhibits a nonzero winding number $W=1$. The spectrum of $\mathcal{H}_\text{F}^{\text{OBC}}$ in the thermodynamic limit is inside the spectrum of $\mathcal{H}_\text{F}^{\text{PBC}}$. (b) The calculated $\text{GBZ}$, which lies inside the unit circle and the two sub-GBZs nearly coincide. (c) Spectrum of $\mathcal{H}_\text{F}^{\text{OBC}}$ for $L=50$ unit cells. The state $|\psi_{j_s}\rangle$ with the maximum SMT is highlighted with a red circle. (d) Spectrum of the effective Hamiltonian $\widetilde{\mathcal{H}}_\text{F}^{\text{OBC}}$ when a controlled potential $\delta V=0.06i\kappa$ is applied to the right end of $\mathcal{H}(z)$ under OBC. Compared with $\mathcal{H}_\text{F}^{\text{OBC}}$, the eigenvalues of most skin modes remain nearly unchanged, except for $|\psi_{j_s}\rangle$ (red circle). (e) Spatial distribution of all eigenstates $|\psi_j\rangle$, where $|\psi_{j_s}\rangle$ is highlighted in red. (f) $|\chi_j|$ of all states, showing that $j_s=50$ corresponds to the skin mode with the maximum SMT.
• Figure 5: SHE of a skin mode via SMT in a Floquet waveguide lattice. (a) Numerical intensity distribution along the propagation direction before adding the end potential perturbation, with input state $|\psi_{j_s}\rangle$, which is not spectrally isolated ($E/\kappa =0.068i$). The result shows that the state does not exhibit self-healing. (b) Propagated intensity distribution after applying the edge perturbation, with the input state $|\psi_{j_s}\rangle$ now being an SHS ($E/\kappa =0.109i$). In both (a) and (b), disturbance in the helical waveguides is indicated by the red rectangle, and only the first 40 waveguides are shown. (c),(d) Deviation of the evolved wavefunction with disturbance, $|\tilde{\Psi}_\text{norm} (z)\rangle$, from that without disturbance, $|\Psi_\text{norm}(z)\rangle$, for (a),(b), defined as $\eta(z)=1-\left |\langle\Psi_\text{norm}(z) \mid \tilde{\Psi}_\text{norm}(z) \rangle\right |$. The region marked in purple indicates the interval during which the disturbance occurs.