Breakup and Recovery of Topological Zero Modes in Finite Non-Hermitian Optical Lattices

The topological edge state (TES) in a one-dimensional optical lattice has exhibited robust field localization or waveguiding against the structural perturbations that would give rise to fault-tolerant photonic integrations. However, the zero mode as a kind of TES usually deviates from the exact zero-energy state in a finite Hermitian lattice due to the coupling between these edge states, which inevitably weaken the topological protection. Here, we first show such a breakup of zero modes in finite Su-Schriffer-Heeger optical lattices and then reveal their recovery by introducing non-Hermitian degeneracies with parity-time ($PT$) symmetry. We carry out experiments in a finite silicon waveguide lattice, where a passive-$PT$ symmetry was implemented with carefully controlled lossy silicon waveguides. The experimental results are fully compatible with the theoretical prediction. Our results show that the topological property of an open system can be tuned by non-Hermitian lattice engineering, which offers a route to enhance the topological protection in a finite system.

Figure 1

Breakup and recovery of zero modes by non-Hermitian modulation. (a) Schematics of the 1D finite non-Hermitian SSH model with $N$ waveguides, where red and blue colors indicate the gain and loss, respectively. (b) Band structure with $N=200$, $c_1=1$, $c_2/c_1=3$, and $W=1$. Gray areas indicate the bulk bands. (c) ${\beta }_A$ and ${\beta }_B$ as a function of $c_2/c_1$ in the Hermitian cases with different $N(c_1=1)$. The red region represents the topologically nontrivial phase and the blue region is the trivial phase. The real (d) and imaginary parts (f) of ${\beta }_A$ and ${\beta }_B$ as a function of $c_2/c_1$ in non-Hermitian cases with different $\gamma$ ($c_1=1$, $N=12$). (e) The eigenmode profiles of exact-zero modes (left) and near-zero modes (right).

Figure 2

Edge mode properties in the non-Hermitian SSH model. (a) $|\mathrm{Re}({\beta }_A-{\beta }_B)|$ and (b) $|\mathrm{Im}({\beta }_A-{\beta }_B)|$ as functions of $N$ and $\gamma /c_1$ with $c_2/c_1=2$ (here $c_1=1$). The black, blue, red, and magenta dots mark the edge states shown in (c)–(e). (c) The real part of eigenvalues as functions of mode number and $\gamma /c_1$, where the gap modes $A$ and $B$ are particularly shown in the zoom-in inset figure. (d) The corresponding imaginary eigenvalue spectra, where the inset depicts the evolutions of real (red curves) and imaginary (blue curves) parts of modes $A$ and $B$ with respect to $\gamma /c_1$. (e) The mode profiles of the five edge states, in which two recovered zero modes (gain and lossy) are both presented. (f) CMT calculated field propagations of exact-$PT$-symmetric mode (the blue case with $\gamma /c_1=0.012$) and two zero modes in the broken-$PT$ phase (the magenta case with $\gamma /c_1=0.05$).

Figure 3

Theoretical model and simulation results. (a) Schematic of the non-Hermitian optical lattice consisting of a dissipative $PT$ silicon waveguide array, the additional chrome (Cr) layer on top of every other silicon waveguide to introduce periodic optical loss modulation. (b) Theoretical mode spectrum of the normalized mode constant as functions of the loss strength ($2\gamma$), where ${\beta }_0^{*}={\beta }_0+i\gamma$ is the averaged propagation constant in the passive system. Two particular cases of $2\gamma =0.005\mu {\mathrm{m}}^{-1}$ (exact-$PT$ phase) and $2\gamma =0.010\mu {\mathrm{m}}^{-1}$ (broken-$PT$ phase) are marked corresponding to the simulation and experimental parameters. Simulated field evolutions of the edge states in (c) exact-$PT$ and (d) broken-$PT$ phases with exactly prepared input excitation states.

Figure 4

Experimental results. (a)–(c) SEM image and enlarged regions of the fabricated structure. (d) CCD recorded optical propagation from input to output through the waveguide lattice. (e) Experimentally detected output intensities (top) and normalized intensity profiles (bottom) of near-zero mode $B$. (f) Corresponding results for the recovered exact-zero mode $B$ (gain mode), where another lossy mode $A$ decays off in propagation. Scale $\mathrm{bar}=10\mu \mathrm{m}$.