Non-Abelian Photonic Lattices: Topological Light Control

• Non-Abelian photonic lattices are engineered optical systems where matrix-valued gauge fields govern light dynamics with noncommuting operations.
• They leverage synthetic dimensions, coupled waveguide arrays, and polarization control to emulate non-Abelian Berry phases and robust topological states.
• Applications include topological quantum computation, robust light-routing, and quantum simulation of gauge theories through scalable photonic platforms.

Non-Abelian photonic lattices constitute a class of engineered optical systems in which the evolution of light is governed by matrix-valued gauge fields, emulating the mathematics and physical consequences of non-Abelian (i.e., noncommutative) symmetry groups. Unlike conventional (Abelian) photonic lattices, these architectures support phenomena such as non-Abelian Berry phases, non-commutative braiding, multiband topological invariants, and complex edge state structures robust against disorder. The realization of non-Abelian physics in photonic lattices leverages internal degrees of freedom—typically polarization, pseudospin, or mode indices—and spatial or synthetic lattice connectivity, enabling the paper and control of topological photonic states and their application in quantum information processing, topological quantum computation, and optical device engineering.

1. Mathematical Framework and Physical Realization

Non-Abelian photonic lattices are grounded in models where the propagation or interaction of photonic modes is subject to matrix-valued link variables or gauge potentials. In a general tight-binding prescription, the Hamiltonian is expressed as

$$H = -J \sum_{\langle m, n \rangle} \Psi_m^\dagger U_{mn} \Psi_n + \text{H.c.},$$

where $\Psi_m$ is a multicomponent field operator (representing, e.g., polarization or orbital pseudospin), and $U_{mn} \in \mathrm{SU}(N)$ is a non-Abelian matrix-valued hopping operator (Cheng et al., 2022, Cheng et al., 1 Jun 2024, Huang et al., 11 Aug 2025).

Realizations employ:

• Synthetic frequency dimensions in ring resonator arrays, in which dynamically controlled electro-optic modulation and polarization rotations engineer the $U_{mn}$ as SU(2) or more general gauge fields (Cheng et al., 2022, Cheng et al., 1 Jun 2024, Wong et al., 18 Nov 2024).
• Waveguide arrays or lattices with spatially modulated couplings, supporting localized modes at engineered defects or domain walls—the Kekulé vortex lattice being a canonical example (Iadecola et al., 2015).
• Programmable spinor lattices using resonators with degenerate pseudospin modes and reconfigurable couplers, allowing implementation of arbitrary unitary rotations and braiding operations (Kim et al., 2 Oct 2024).

The non-commutativity $[U_{mn}, U_{pq}] \ne 0$ for different links, or of the underlying gauge potentials $[A_\mu, A_\nu] \ne 0$, is essential for non-Abelian physics. Physically, this translates to optical paths or operations where the final state depends not just on the geometric path in the parameter space, but also on the sequence of applied operations.

2. Non-Abelian Berry Phases and Braiding

A central feature of non-Abelian photonic lattices is the existence of geometric (Berry) phases that are operator- or matrix-valued, reflecting evolution within a degenerate subspace. For light injected into "topological guided modes" at vortex defects in Kekulé-distorted lattices, adiabatic braiding of these defects leads to non-Abelian Berry phases described by unitary braid matrices. The generic transformation for such a process is given by

$$\tau_i = \exp\left[\frac{\pi}{2} (b_{0,i+1}^\dagger b_{0,i} - b_{0,i}^\dagger b_{0,i+1})\right],$$

acting on the creation operators of vortex zero-modes, with the composition of such $\tau_i$ representing a noncommutative braid group (Iadecola et al., 2015).

Photonic chip architectures enable controlled experimental realization of such multi-mode braiding, where the sequential application of non-Abelian gates (for example, $G_1, G_2$) leads to genuinely order-dependent (non-commutative) outcomes, as directly measured in both classical and single-photon interference experiments (Zhang et al., 2021).

Shortcuts to adiabaticity further enable these non-Abelian gate operations—in particular, Pauli X, Y, Z gates and their composition—on silicon photonic chips with a spatial footprint three orders of magnitude smaller than previous waveguide-based devices (Song et al., 9 Oct 2024).

3. Lattice Gauge Fields, Synthetic Dimensions, and Spin-Orbit Coupling

Non-Abelian lattice gauge fields are engineered by mapping photonic internal degrees of freedom (e.g., polarization) onto pseudospin and creating SU(2) link variables in photonic lattices. Synthetic frequency dimensions are realized in ring resonator arrays by dynamic modulation and polarization-multiplexed phase operations: $$U_m = \exp[i \beta_m \sigma_z] \cdot \exp[i \Omega_m \sigma_x]$$ with $\sigma_{x,z}$ denoting Pauli matrices acting on the polarization or spin basis (Cheng et al., 2022).

These link variables naturally generalize the Harper–Hofstadter model for quantum Hall systems to a non-Abelian context, producing matrix-valued fluxes and giving rise to features such as:

• Spin-orbit-coupled Hofstadter butterfly spectra, with polarization-momentum-locked edge states (Cheng et al., 2022).
• Dirac cones at time-reversal-invariant momenta in lattices with non-commuting gauge potentials $A_x \neq A_y$, observable via full polarization-state tomography (Cheng et al., 1 Jun 2024).
• Synthetic non-Abelian electric fields derived from Yang–Mills theory, realized via commutators $[V, A_x]$ where $V$ and $A_x$ do not commute, leading to phenomena such as photonic Zitterbewegung and mixed Rashba–Dresselhaus spin–orbit coupling (Wong et al., 18 Nov 2024).

In dissipative photonic lattices, matrix-valued Wilson line operators describe the system's evolution, and noncommutativity between these lines gives rise to intrinsically non-Abelian geometric effects even in the presence of loss or gain (Parto et al., 2022).

4. Topological Invariants, Edge States, and Bulk-Interface Structure

Non-Abelian photonic lattices support topological invariants that go beyond Abelian Berry phases and Chern numbers:

• The Wilczek–Zee holonomy generalizes the Berry phase to degenerate bands, captured by non-Abelian field strengths

$$\mathcal{F}^{\nu}_{kz} = \partial_k \Gamma^z_\nu - \partial_z \Gamma^k_\nu + i[\Gamma^z_\nu, \Gamma^k_\nu]$$

where the noncommutativity signals a gauge structure for the underlying band manifold (Brosco et al., 2020).

• N-band Hopf invariants, computed using the non-Abelian third Chern–Simons form,

$$n_H = \int_{\text{BZ}} \frac{d^3k}{8\pi^2} \mathrm{Tr}\left\{ \mathcal{A} \cdot \left( \nabla_k \times \mathcal{A} - \frac{2i}{3} \mathcal{A} \times \mathcal{A} \right) \right\},$$

classify topological phases in microring lattices beyond the standard Chern framework (Leng et al., 2022).

• The quaternionic "frame charge" formalism for nodal lines in PT-symmetric photonic crystals gives rise to non-integer, non-Abelian topological charges that flip sign upon the braiding of degeneracy lines, in contrast to integer-valued (Abelian) charges (Wang et al., 2022).

Interfaces between Abelian bulks can host non-Abelian topological phenomena: programmatically switching the rotation axis of the coupling in programmable spinor lattices produces interfaces supporting edge states protected not only by global topology but also by the local noncommutative (non-Abelian) gauge field structure (Kim et al., 2 Oct 2024).

5. Non-Hermitian Effects, Exceptional Points, and Braid Monopoles

Non-Hermitian extensions of non-Abelian photonic lattices—incorporating complex hopping, gain/loss, and asymmetric couplings—produce phenomena where the interplay of noncommutativity and band structure non-Hermiticity induces new classes of topological protection:

• The braiding of complex eigenenergies yields "nested Hopf-links" in the complex energy plane. These braids remain robust below a critical value of non-Hermiticity parameter $\gamma$, but at the exceptional point (EP) a topological phase transition removes the linked structure, opening an imaginary spectral gap (Gupta, 27 Jun 2025).
• Open boundary spectra display "conjoint open-arcs" with a purely dipole skin effect—i.e., all eigenstates are localized at the system edge with no extended states—a feature absent in simpler non-Abelian or Abelian systems (Gupta, 27 Jun 2025).
• The realization of a monopole degeneracy at a third-order exceptional point (EP3) demonstrates non-Abelian braid topology, as encoded by the non-commuting braid group word $$B_M = \sigma_1 \sigma_2 \sigma_1^{-1} \sigma_2^{-1}$$, protected from the conventional Hermitian doubling theorem (Wang et al., 10 Oct 2024).
• Non-Abelian fusion rules in these systems manifest as path-dependent conjugacy classes of braiding operations around exceptional points, with implications for error-resilient information processing (Wang et al., 10 Oct 2024).

6. Light–Matter Interactions and Quantum Electrodynamics

The embedding of quantum emitters (two-level systems) in non-Abelian photonic lattices broadens the quantum optical landscape. Calculations reveal:

• Chiral photon emission and vortex states arise from the spin-momentum-locked band structure, with the emission direction and orbital angular momentum of output photons dependent on the emitter pseudospin (Huang et al., 11 Aug 2025).
• The Landau–Jaynes–Cummings model in the presence of both Abelian and non-Abelian fields results in spin-polarized and squeezed Landau polaritons, with Rabi frequencies tunable via gauge field parameters and Landau level index (Huang et al., 11 Aug 2025).
• Nonsymmorphic crystalline symmetry yields real-space staggered phases, leading to position-dependent Purcell enhancement or suppression in multi-emitter arrays, enabling engineering of chiral, highly correlated quantum networks (Huang et al., 11 Aug 2025).

7. Applications and Prospects

Non-Abelian photonic lattices find application as:

• Testbeds for topological quantum computation, where non-Abelian braiding of photonic modes can be mapped to quantum logic gate operations (Y, X, Z gates and beyond) (Zhang et al., 2021, Song et al., 9 Oct 2024).
• Platforms for robust control of light routing, information processing, and (potentially) topological quantum error correction (Kim et al., 2 Oct 2024, Song et al., 9 Oct 2024).
• Experimental systems for exploring dissipative topological phases, non-Abelian skin effects, and edge-state lasing (Parto et al., 2022, Gupta, 27 Jun 2025).
• Synthetic quantum simulators of high-energy gauge theory phenomena, such as Wilson loop dynamics and non-Abelian Aharonov–Bohm caging (Li et al., 2020, Pang et al., 4 Dec 2024).

Unifying the design space of synthetic frequency dimensions, programmable coupler lattices, and interaction-enabled QED, these systems enable precise control of internal (pseudospin) and external (lattice, frequency, spatial) degrees of freedom in optical fields. The flexibility and tunability afforded by integrated photonics platforms, as well as compatibility with CMOS fabrication (Song et al., 9 Oct 2024), make non-Abelian photonic lattices promising candidates for scalable next-generation quantum and classical optical systems.