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Optica

Topological quantum walk in synthetic non-Abelian gauge fields with photonic mesh lattices

Zehai Pang, Omar Abdelghani, Marin Soljacic A, Yi Yang

Published Aug 11, 2025• 2 citations
Publisher

Abstract

Floquet systems like quantum walks can exhibit anomalous topological boundary modes that have no stationary counterpart. Although the topological properties of synthetic gauge fields have been explored in quantum walks, these studies have so far focused on Abelian gauge fields. We theoretically introduce synthetic non-Abelian gauge fields for topological quantum walks and study their topological consequences. The photonic mesh lattice configuration is generalized with polarization multiplexing to achieve a four-dimensional Hilbert space, based on which we provide photonic building blocks for realizing various quantum walks in non-Abelian gauge fields. It is found that SU(2) gauge fields can lead to Peierls substitution in both momenta and quasienergy. In one and two dimensions, we describe detailed photonic setups to realize topological quantum walk protocols whose Floquet winding numbers and Rudner–Lindner–Berg–Levin invariants can be effectively controlled by the gauge fields. Finally, we show how non-Abelian gauge fields facilitate convenient simulation of entanglement in conjunction with polarization-dependent and spatial-mode-dependent coin operations. Our results shed light on the study of synthetic non-Abelian gauge fields in photonic Floquet systems and hold implications for time-multiplexed optical networks.

DOI
10.1364/optica.567878

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Floquet systems like quantum walks can exhibit anomalous topological boundary modes that have no stationary counterpart. Although the topological properties of synthetic gauge fields have been explored in quantum walks, these studies have so far focused on Abelian gauge fields. We theoretically introduce synthetic non-Abelian gauge fields for topological quantum walks and study their topological consequences. The photonic mesh lattice configuration is generalized with polarization multiplexing to achieve a four-dimensional Hilbert space, based on which we provide photonic building blocks for realizing various quantum walks in non-Abelian gauge fields. It is found that SU(2) gauge fields can lead to Peierls substitution in both momenta and quasienergy. In one and two dimensions, we describe detailed photonic setups to realize topological quantum walk protocols whose Floquet winding numbers and Rudner–Lindner–Berg–Levin invariants can be effectively controlled by the gauge fields. Finally, we show how non-Abelian gauge fields facilitate convenient simulation of entanglement in conjunction with polarization-dependent and spatial-mode-dependent coin operations. Our results shed light on the study of synthetic non-Abelian gauge fields in photonic Floquet systems and hold implications for time-multiplexed optical networks.
We theoretically introduce synthetic non-Abelian gauge fields for topological quantum walks. The photonic mesh lattice configuration is generalized with polarization multiplexing to achieve a four-dimensional Hilbert space, based on which we provide photonic building blocks for realizing various quantum walks in non-Abelian gauge fields. It is found that SU(2) gauge fields can lead to Peierls substitution in both momenta and quasienergy. In one and two dimensions, we describe detailed photonic setups to realize topological quantum walk protocols whose Floquet winding numbers and Rudner-Lindner-Berg-Levin invariants can be effectively controlled by the gauge fields. Finally, we show how non-Abelian gauge fields facilitate convenient simulation of entanglement in conjunction with polarization-dependent and spatial-mode-dependent coin operations. Our results shed light on the study of synthetic non-Abelian gauge fields in photonic Floquet systems.
There is great interest in designing photonic devices capable of disorder-resistant transport and information processing. In this work we propose to exploit three-dimensional integrated photonic circuits in order to realize two-dimensional discrete-time quantum walks in a background synthetic gauge field. The gauge fields are generated by introducing the appropriate phase shifts between waveguides. Polarization-independent phase shifts lead to an Abelian or magnetic field, a case we describe in detail. We find that, in the disordered case, the magnetic field enhances transport due to the presence of topologically protected chiral edge states that do not localize. Polarization-dependent phase shifts lead to effective non-Abelian gauge fields, which could be adopted to realize Rashba-like quantum walks with spin-orbit coupling. Our work introduces a flexible platform for the experimental study of multiparticle quantum walks in the presence of synthetic gauge fields, which paves the way towards topologically robust transport of many-body states of photons.
We propose a non-Hermitian Floquet photonic lattice formed by a physical resonator coordinate and a synthetic frequency coordinate. A two-step modulation protocol realizes a chiral walk in this physical-synthetic plane, with a real synthetic flux controlling loop interference and imaginary gauge fields controlling non-reciprocal envelopes. We show that anomalous corner pairs at quasienergies zero and π/Tπ/Tπ/T exhibit three distinct layers of physics. A non-Bloch higher-order construction predicts whether the 0/π0/π0/π corner pair exists under open boundaries. The imaginary gauge fields select where the right eigenmodes accumulate. The real flux controls the local interference matrix element that determines whether the doubled-period optical response is visible. As a result, the same topological coexistence sector can be bright, skin-dark, or flux-dark in a local optical measurement. We further show that the complex gauge can tune an exceptional point of the two-period corner propagator. At this point the anomalous response keeps its doubled-period sign alternation, but its envelope becomes algebraic because of a Jordan block. These results provide a photonic route to separate topological existence, skin-selected localization, optical visibility, and defective two-period dynamics in a non-Hermitian synthetic dimension.
Temporal multiplexing provides an efficient and scalable approach to realize a quantum random walk with photons that can exhibit topological properties. But two-dimensional time-multiplexed topological quantum walks studied so far have relied on generalizations of the Su-Shreiffer-Heeger model with no synthetic gauge field. In this work, we demonstrate a two-dimensional topological quantum random walk where the nontrivial topology is due to the presence of a synthetic gauge field. We show that the synthetic gauge field leads to the appearance of multiple band gaps and, consequently, a spatial confinement of the quantum walk distribution. Moreover, we demonstrate topological edge states at an interface between domains with opposite synthetic fields. Our results expand the range of Hamiltonians that can be simulated using photonic quantum walks.
The concept of multi-gap topology has recently been shown to give rise to uncharted phases beyond conventional single-gap classifications. These phases relate to band nodes with non-Abelian quaternion charges and momentum-space braiding processes characterized by new invariants such as paradigmatic Euler class, phenomena that intrinsically require at least two spatial dimensions. Extending such phases into the non-equilibrium regime is predicted to unlock even richer multi-gap topologies beyond static settings, yet their experimental realization has remained elusive due to the stringent requirements on dimensionality, symmetry, and dynamical control. Here, we theoretically demonstrate and, for the first time, experimentally realize two-dimensional (2D) Floquet non-Abelian band topology in photonic scattering networks. Within this platform, we uncover a sequence of topological phenomena unique to 2D multi-gap systems far from equilibrium, including anomalous multi-gap phases interconnected by band nodes, Floquet Euler transfer, gapped phases with anomalous Dirac string configurations, and Floquet-induced non-Abelian braiding of band nodes. In addition, we observe Floquet-periodic anomalous edge states across multiple gaps, providing experimental signatures of these sought-after 2D multi-gap Floquet topological phases. Our results establish photonic scattering networks as a practical and versatile route to non-Abelian Floquet systems, opening avenues for dynamical topological physics with braiding capability and robust photonic functionalities.