Programmable lattices for non-Abelian topological photonics and braiding

Technical Summary: Programmable Lattices for Non-Abelian Topological Photonics and Braiding

Problem Statement
While programmable photonic circuits (PPCs) have successfully established reconfigurable universal SU(2) gates for high-level wave manipulation and matrix calculations, extending these capabilities to non-Abelian physics remains a significant challenge. Non-Abelian systems require matrix-valued gauge fields within noncommutative unitary groups U(N>1), where the noncommutative nature of internal symmetries is central. Previous photonic implementations of non-Abelian gauge fields have relied on static platforms or specific reconfigurable setups using anisotropic materials, metamolecules, or frequency-synthetic dimensions. However, a versatile, reconfigurable, and lattice-compatible building block capable of emulating both Abelian and non-Abelian topological phenomena, particularly those involving noncommutative operations at interfaces, has been lacking.

Methodology
The authors propose a programmable photonic building block designed to realize reconfigurable U(2) gauge fields. The core component is a travelling-wave ring resonator lattice where each resonator supports degenerate pseudospin resonances (counter-clockwise and clockwise), forming a pseudospinor state.

• Building Block Design: The fundamental unit consists of two travelling-wave resonators coupled via a nonreciprocal loop coupler. This coupler integrates an SU(2) gate and global phase shifters. Crucially, the design employs a nonreciprocal phase shifter (NRPS) implemented using a cerium-substituted yttrium iron garnet (Ce:YIG) silicon waveguide. This allows for the tuning of local phase shifts ($\xi_L$) via an external magnetic field, which is the critical parameter for coupling counter-clockwise and clockwise resonances to achieve non-Abelian U(2) gauge fields.
• Hamiltonian Formulation: The lattice is governed by a tight-binding Hamiltonian with matrix-valued gauge fields. The link variable $U_{mn}$ is tailored by local phase shifts ($\xi_L, \eta_L$) and global shifts, enabling complete rotations around the y- and z-axes of the spinor Bloch sphere.
• Simulation and Analysis: The authors utilize finite-difference-frequency-domain (FDFD) and finite-difference-time-domain (FDTD) methods (via Tidy3D) to design components and analyze eigenmodes. They theoretically investigate the system by calculating Hofstadter butterflies for various loop operators and analyzing band structures using supercell configurations to model interfaces.

Key Contributions and Results

1. Isospectral Abelian Topological Lattices:
   The authors demonstrate that their platform can emulate an isospectral family of Abelian topological phenomena, specifically the Quantum Hall Effect (QHE) and Quantum Spin Hall Effect (QSHE), by programming the distribution of coupler phase shifts.

   • By setting the loop operator $K$ to specific forms (e.g., $K_0$, $K_y$, $K_z$), they realize different eigenspinor bases.
   • They show that while the QHE breaks time-reversal symmetry with identical spin gap Chern numbers for both pseudospins, the QSHE preserves global time-reversal symmetry with opposite signs for each pseudospin.
   • This establishes a single platform capable of dynamically engineering eigenspinor bases and time-reversal symmetry properties.

2. Non-Abelian Topological Interfaces:
   A primary contribution is the introduction and demonstration of "non-Abelian interfaces." These are interfaces formed between two Abelian topological bulks (e.g., a lattice with loop operator $K_y$ adjacent to one with $K_z$).

   • Noncommutativity: Although the bulk regions are Abelian, the interface exhibits non-Abelian physics because the loop operators $K_y$ and $K_z$ do not commute ($[\sigma_y, \sigma_z] \neq 0$).
   • Edge State Hybridization: Unlike standard Abelian interfaces where edge states are purely topologically protected, these non-Abelian interfaces reveal the coexistence of topologically nontrivial edge states and topologically trivial hybridizations. This leads to the reopening of bandgaps, a phenomenon unique to non-Abelian interface physics.
   • Topological Trivial Engineering: The authors show that topologically protected edge states can be engineered even when the bulks are topologically trivial in specific bases, provided the interface distribution is non-Abelian.

3. Non-Abelian Resonant Braiding:
   The paper demonstrates the classical emulation of non-Abelian braiding operations for pseudospin observables.

   • Braid Group $B_3$: By constructing a 1D coupled-resonator lattice, the authors map the 2+1 space-time dimensions of non-Abelian anyons to the 2D Bloch sphere surface and 1D resonant coupling.
   • Generators and Relations: Using rotation operations $U_y$ and $U_z$ as generators, they verify the criteria for the braid group $B_3$, including the non-Abelian condition ($U_y U_z \neq U_z U_y$) and the Yang–Baxter relation ($U_y U_z U_y = U_z U_y U_z$).
   • Experimental Realization: Transmission spectra confirm that these relations hold across the entire spectrum, with perfect conservation of strands (spin observables) occurring at resonant-tunnelling frequencies.

Significance
The paper claims to provide a foundational building block for non-Abelian and programmable topological photonics. Its significance lies in:

• Versatility: Offering a reconfigurable testbed for a wide class of both Abelian and non-Abelian topological phenomena on a single platform.
• New Physics Domain: Extending non-Abelian topological photonics into the realm of interface physics, revealing unique phenomena like gap reopening and hybrid edge states that are distinct from conventional bulk-boundary correspondence.
• Programmability: Enabling dynamic control over time-reversal symmetry, bulk-edge configurations, and braiding operations through simple phase shifter adjustments.
• Braiding Emulation: Providing a resonant, discretized, and spectral realization of braid groups for pseudospin observables, contrasting with previous approaches based on propagating, adiabatic, and spatial-mode realizations.

The authors note that while the current design utilizes Ce:YIG for nonreciprocity, time-varying modulations could offer magnetic-free alternatives in the future. They also suggest that extending this research to point-like non-Abelian configurations and exploring the complex spectra of non-Abelian interfaces are potential future directions.