Topological edge states in curved zigzag superlattices in nonlinear exciton-polaritons

TL;DR

This work addresses the generation and control of topological edge states in a 1D zigzag SSH-like lattice built from two staggered sublattices. By combining curvature as a tunable geometric parameter with nonlinear exciton-polariton dynamics described by a driven-dissipative Gross-Pitaevskii equation, the authors show how edge states can be enhanced, suppressed, or made to coexist within the same band gaps, including through nonlinear resonant pumping. Key findings include curvature-induced modulation of intra-lattice couplings that reshapes edge-state existence regions and the ability to realize coexistence and switching of edge states via nonlinearity, with exciton-polaritons providing a practical platform. The work expands the design space for complex topological photonic devices and suggests routes for optical imprinting and flexible-material implementations to tune curvature-driven topological phenomena.

Abstract

Zigzag chains allow for the formation of topological edge states. Several distinct chain architectures have been developed for this purpose. Here, we report a zigzag superlattice, containing two staggered sub-lattices, that supports multiple edge states, including higher-order modes. In such lattices, the intra- and intercell coupling is imbalanced by the tunneling effect of the eigenstates or deformation of the higher-order modes. We demonstrate that by arranging the zigzag superlattice into a curved shape, some of the edge states transition into bulk states as the curvature of the lattice increases, while some bulk states become more localized towards edge states. The reason is that a curved superlattice strengthens the intra-lattice coupling of the inner sub-lattice due to the separation reduction of the potential wells. %which, on the one hand, hinders the tunneling of the eigenstate to the outer sub-lattice and hence weakens the edge states formed in the inner sub-lattice. On the other hand, it can isolate more edge states formed in the outer sub-lattice, because of the induced more intensive deformation of the higher-order modes in the inner sub-lattice. We also show that some bulk states at larger curvatures can be transformed into edge states by a repulsive nonlinearity, which also enables the coexistence of different edge states. As a specific and ideal platform for realizing such topological superlattices we explore exciton-polaritons in semiconductor microcavities with their strong nonlinearity and possibility for optical excitation and control. Our work introduces an additional dimension for the design of complex topological lattices and functional photonic devices.

Figures (5)

• Figure 1: Lattice structures and eigenstates. (a) Distribution of a potential chain, Lattice-1, containing 15 identical potential wells with a diameter of 4 $\mu$m and a separation, center to center, of 4 $\mu$m. (b) A similar chain with a smaller well diameter of 2.8 $\mu$m, Lattice-2. (c) A superlattice composed of staggered Lattice-1 and Lattice-2 with the separation of the nearest larger and smaller potential wells of 3.4 $\mu$m, forming a perfect zigzag shape. The solid and dashed lines indicate the intracell and intercell coupling directions, respectively. (d,e) Dependence of the first four energy bands on the potential depth of Lattice-1 and Lattice-2, respectively. The inset in (d) shows the first 90 eigenvalues at the potential depth of $V_1$=5 meV, and the inset in (e) shows the first 90 eigenvalues at the potential depth of $V_2$=8 meV. (f) Dependence of the energy bands of the zigzag superlattice in (c) on the difference of the potential depths with $V_1$=5 meV fixed. Black lines represent bulk states, while colored lines represent edge states. The insets are the enlarged view of the corresponding dashed boxes. (g-i) Amplitude (upper) and phase (lower) profiles of the eigenstates marked in (d-f). The circles in (g,h) indicate an individual potential well in each lattice.
• Figure 2: Edge eigenstates in standard zigzag superlattices. (a) Dependence of edge states on the difference of the potential depths of the two sub-lattices with $V_1$=5 meV fixed, extracted from Fig. \ref{['fig:1']}(c). Amplitude distribution of a (b) R1, (c) R2, (d) R3, (f) R4, and (g) L2 edge state with the lattice structure marked on top. (e) Phase of the R1 edge state in (b) with the last three potential wells marked on top.
• Figure 3: Edge states in curved superlattices. (a) Examples of differently curved chains with $V_1$=5 meV and $V_2$=7 meV. Existence areas of the (b) R1 and R2, (c) R3, (d) L1, and (e) L2 edge states in relation to the potential depth difference (here $V_1$=5 meV) and the curvature of the chain. The colored areas represent edge states, while the white areas represent bulk states. The colorbar indicates the ratio of the peak amplitude in the second well from the edge (in the same sub-lattice) and the peak amplitude in the edge well, i.e., $|\Psi^\textup{Lattice-1(2)}_{\textup{edge}\pm1}|^\textup{max}/|\Psi^\textup{Lattice-1(2)}_{\textup{edge}}|^\textup{max}$. The smaller the ration, the more prominent the edge state. (f) Normalized peak amplitudes of all potential wells in Lattice-1, indicated by the dashed curve in (a), of the bulk states selected along the dashed line in (d) at different curvatures.
• Figure 4: Edge states enhanced by the intra-lattice coupling. Amplitude distributions of (a) the state in a zigzag chain with the curvature of 0.05, 15 wells, $V_1$=5 meV, and $V_2$=7.5 meV, corresponding to the state marked by the black point in Fig. \ref{['fig:3']}(d), (b) the state in a similar chain as in (a) but more, 31, potential wells, (c) the state in a similar chain as in (a) but a increased well diameter, 3 $\mu$m, of the potential wells in Lattice-2, and (d) the state in a similar chain as in (c) but a larger potential depth, $V_2$=7.75 meV, of the potential wells in Lattice-2.
• Figure 5: Edge states in the nonlinear regime. (a) Existence area of the R1 edge state in the nonlinear regime, i.e., under resonant excitation with $E_0=10.234~\mu$eV $\mu$m$^{-1}$, based on the linear results in Fig. \ref{['fig:3']}(b). (b) Density ($|\Psi|^2$, in $\mu$m$^{-2}$) distribution of an example bulk state, corresponding to the point in (a). (c-e) Density distributions of the coexisting L2 and R3 edge states at standard superlattices (left column), i.e., the curvature is 0 as shown in Fig. \ref{['fig:1']}(c), and at curved superlattices (right column) with the curvature of 0.06, referring to the lattice structures in Fig. \ref{['fig:3']}(a). The potential depths are (c) $V_2=7.9$ meV, (d) $V_2=8$ meV, and (e) $V_2=8.1$ meV. Here, the excitation amplitude is $E_0=66.522~\mu$eV $\mu$m$^{-1}$. In all cases, $V_1=5$ meV.