Non-Abelian interference of topological edge states

TL;DR

The paper addresses realizing non-Abelian interference and entanglement of topological edge states using dual symmetry protection in periodically driven coupled SSH chains. It develops a double-chain platform with inversion symmetry and time-dependent interchain symmetry to achieve permutation of edge-state pairs and symmetry-protected adiabatic transport, governed by a dynamical phase φ_d. Extending to three chains, it demonstrates non-Abelian permutation among three edge-state pairs and realizes non-Abelian topological transport and Hong-Ou-Mandel interference that generate spatial NOON states with high fidelity. The work provides a robust, symmetry-based route to non-Abelian topology for quantum information tasks, with experimental viability in photonic waveguide arrays and potential extensions to bulk states and spin degrees of freedom.

Abstract

Topological boundary states exhibit distinctive properties, including unidirectional propagation and noise robustness, which hold significant potential for advancing the performance of quantum science and technology. Here, we demonstrate the implementation of non-Abelian quantum interference and entanglement generation, protected by dual symmetries (time-independent inversion and time-dependent interchain), in coupled Su-Schrieffer-Heeger chains. Specifically, in a multichain system, we first achieve tunable topological transfer of a single particle, where the destination chain is selected by the permutation sequence. We then extend this to two particles, observing a non-Abelian Hong-Ou-Mandel interference that generates spatially entangled NOON states whose properties are dictated by the permutation sequence. Our work establishes an alternative pathway for exploring non-Abelian topology applied to quantum science and technology, enabled by the unique protection of time-dependent symmetry.

Figures (8)

• Figure 1: Schematic diagram of double-coupled SSH chains $A$ and $B$. $v$ and $w$ denote the alternate intrachain hopping, $J$ represent the interchain hopping, and $2\Delta$ are on-site energy offsets.
• Figure 2: permutation of two pairs of topological edge states. (a) Energy spectrum of the double-coupled SSH chains. Black solid lines represent the bulk states, and colored lines represent pairs of edge states. (b) Two branches of the expectation value $\langle\bm{S}_D\rangle$. (c) Abelian braiding processes of two pairs of edge states in the ($t$, $\text{Im}[\langle\bm{S}_D\rangle]$, $E$) space, corresponding to the braiding operators $\tau^{-1}_{1}$. In all panels, the color denotes the argument of the expectation value $\langle\bm{S}_D\rangle$. The parameters set as $L=14$, $w=1$, $v_{0}=0.5$, and $\kappa=0.2$.
• Figure 3: Symmetry-protected adiabatic evolution. (a) Expectation values $\langle\bm{S}_D\rangle$ (colored dots) and eigenvalues $S_{D,\text{I}}$ (colored lines) during adiabatic evolution. The color indicates the argument of $\langle\bm{S}_D\rangle$ or $S_{D,\text{I}}$. (b) Conservation of parity during adiabatic evolution. (c) Density distribution $\langle\hat{n}_{j}\rangle$ as a function of evolution time $t$. (d) Distribution of the final state at the two ends ($|1_{B}\rangle$ and $|L_{B}\rangle$) of chain $B$ versus $\phi_{d}\in[\pi,3\pi]$. The parameters set as $L=14$, $w=1$, $v_{0}=0.5$, $\kappa=0.2$. For panels (a), (b), and (c), $T=1332$ (corresponding to $\phi_{d}=1.5\pi$).
• Figure 4: Schematic diagram of triple-coupled SSH chains $A$, $B$ and $C$. $v$ and $w$ denote the alternate intrachain hopping. $J_{AB}$, $J_{AC}$, and $J_{BC}$ represent the interchain hopping amplitudes between chain $A$ and $B$, $A$ and $C$, and $B$ and $C$, respectively, while $\Delta_{A}$, $\Delta_{B}$, and $\Delta_{C}$ denote the on-site energies of chain $A$, $B$, and $C$, respectively.
• Figure 5: Non-Abelian permutation of three pairs of edge states. (a) Energy spectrum of edge states for the triplet-coupled SSH chains. (b) Two branches of the expectation value $\langle\bm{S}_T\rangle$. (c), (d) Non-Abelian braiding processes of three pairs of edge states in the ($t$, $\text{Im}[\langle\bm{S}_T\rangle]$, $E$) and ($t$, $\text{Im}[\langle\bm{S^*}_T\rangle]$, $E$) space, corresponding to the braiding operators $\tau^{-1}_{1}\tau^{-1}_{2}$ (panel (c)) and $\tau^{-1}_{2}\tau^{-1}_{1}$ (panel (d)). In all panels, the color indicates the argument of the expectation value $\langle\bm{S}_T\rangle$ or $\langle\bm{S^*}_T\rangle$. The parameters set as $L=14$, $w=1$, $v_{0}=0.5$, and $\eta=0.2$.