Quantized non-Abelian helicity of flat bands in 2D Floquet topological photonic insulators

Abstract

Flat-band states in topological systems provide a unique platform for investigating strongly correlated phenomena and many body physics. However, in 2D static tight-binding systems, perfectly flat bands can only exist in the topologically trivial phase, as characterized by a zero Chern number. Here we show that by introducing periodic driving into a 2D photonic Lieb lattice composed of coupled microring resonators, the resulting Floquet topological insulator can host perfectly flat bands with nontrivial topology. In particular, by tracking the evolution of the flat-band modes over each cycle, we show that the non-Abelian displacements of the flat-band modes are characterized by a nontrivial quantized helicity even though the quasi-energy bands have zero Chern number. The helical motion of the flat-band modes can be described by a braiding of the world lines of their trajectories, with a nontrivial winding number directly connected to the helicity. We also propose a scheme to experimentally measure the quantized non-Abelian helicity in a microring lattice subject to a synthetic magnetic field. These results suggest that Floquet topological photonic insulators based on coupled microring resonators can provide a versatile platform for investigating non-Abelian topological physics and strongly correlated phenomena in photonic flat-band systems.

Figures (6)

• Figure 1: (a) Schematic of the Floquet-Lieb insulator realized using coupled microring resonators. The dashed red lines indicate a unit cell consisting of three microrings A, B, C. The arrows in the rings show light circulating around each ring and coupling to neighbor resonators. (b) Coupling sequence of the FLI lattice with each dot representing a site microring waveguide. (c) Quasienergy band diagram of the FLI with coupling angle $\theta = 0.45 \pi$. The Chern numbers ($C$) of the bands and winding numbers ($W$) of the band gaps are also shown. (d) Quasienergy spectrum of the FLI (blue color) projected over all $(k_x, k_y)$ values vs. the coupling angle. Grey regions indicate trivial bandgaps with winding number $W = 0$; white regions indicate nontrivial bandgaps with $W = 1$.
• Figure 2: (a) Initial displacements $x_{\mathrm{c},n}(0) = y_{\mathrm{c},n}(0)$ of the Wannier center of Floquet mode $|\Phi_n\rangle$ vs. the coupling angle $\theta$. The sums of the displacements in $x$ and $y$ are indicated by the 2D Zak phases $\Theta_x$ and $\Theta_y$, respectively. (b) and (c) Trajectories $\Delta x_{\mathrm{c},n}(z)$ (upper panel) and $\Delta y_{\mathrm{c},n}(z)$ (lower panel) of the Wannier center vs. $z$ over one period for (b) flat-band mode ($|\Psi_1\rangle$) and (c) dispersive modes ($|\Psi_2\rangle$ and $|\Psi_3\rangle$) for various coupling angles.
• Figure 3: Helicity $\mathcal{H}$ of the Floquet modes of an FLI lattice as function of the coupling angle. The helicity contributions of the flat-band mode $(\mathcal{H}_1)$ and the two dispersive modes $(\mathcal{H}_2 = \mathcal{H}_3)$ are also shown.
• Figure 4: An FLI lattice with $N_x \times N_y$ square unit cells subject to a synthetic magnetic flux $\alpha$ generated by a coupling phase gradient $\phi_m = \pi\alpha m$ along the $x$ direction.
• Figure 5: Simulated transmission spectra over one FSR of an FLI lattice subject to different values of applied magnetic flux $\alpha$ for (a) perfect coupling case ($\theta = 0.5\pi$) and (b) $\theta = 0.48\pi$. (c) Shifts in resonance frequencies (in terms microring roundtrip phase) of the Floquet modes $n$ as functions of the applied flux $\alpha$ for FLI lattice with $\theta = 0.5\pi$ (blue line) and $\theta = 0.48\pi$ (red dots). The red dashed line is the sum of the roundtrip phase shifts of the three modes.