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arXiv:2412.03043

Topological quantum walk in synthetic non-Abelian gauge fields

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

Published Dec 4, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2412.03043

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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.
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.
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.
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.
Theoretical analysis indicates that a spin qubit in a parabolic quantum wire, when driven by a bichromatic field, generates a confinement-tunable non-Abelian synthetic gauge field that gives rise to novel Floquet topological phenomena. In the low-confinement regime, the system exhibits topological Landau-Zener (LZ) transitions, chiral Landau-Zener-Stu¨\ddot{u}u¨ckelberg-Majorana (LZSM) interference, and non-Abelian geometric phases around diabolical points. These features enable holonomic quantum computation, as well as unconventional Floquet Bloch oscillations and fractal spectra. In contrast, in the strong confinement regime, narrow avoided crossings significantly suppress cross-coupling, thereby insulating the qubit from multi-photon LZSM interference. Quantitative Floquet-Lindblad analysis shows that the optimized gate times, determined by the biharmonic driving period and non-Abelian parameter-space loops, are sufficiently short to surpass primary environmental decoherence channels. When applied to high-mobility GaAs/AlGaAs\text{GaAs/AlGaAs}GaAs/AlGaAs core-shell nanowires, this approach converts spatial drives into robust non-Abelian gauge fields, positioning quantum wire systems as a versatile and scalable platform for fault-tolerant quantum information processing.
Aharonov-Bohm (AB) caging is a complete localization phenomenon in two-dimensional lattices due to destructive interference induced by the background gauge fields. However, current investigations of AB caging are mostly restricted to the Abelian gauge field case, and the observation of AB caging under non-Abelian gauge fields in a quantum system still remains elusive. Here, we report experimental realization of tunable synthetic non-Abelian SU(2) gauge fields in a rhombic lattice, engineered within the synthetic dimensions of a vibrating trapped ion with multiple levels. We realize AB caging under both Abelian and non-Abelian gauge fields and systematically investigate the distinctive transport properties of the non-Abelian case. In particular, we observe typical emergent quantum dynamics unique to non-Abelian AB caging, including initial-state-dependent dynamics, second-order effects, and asymmetric caging behavior. These observations demonstrate the trapped ion system as a powerful platform for simulating emergent phenomena in high-dimensional quantum systems with exotic synthetic gauge fields.