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Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions

Nature volume 637, pages 52–56 (2025)Cite this article

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Abstract

Non-Abelian gauge fields1 provide a conceptual framework to describe particles having spins, underlying many phenomena in electrodynamics, condensed-matter physics2,3 and particle physics4,5. Lattice models6 of non-Abelian gauge fields allow us to understand their physical implications in extended systems. The theoretical importance of non-Abelian lattice gauge fields motivates their experimental synthesis and explorations7,8,9. Photons are fundamental particles for which artificial gauge fields can be synthesized10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30, yet the demonstration of non-Abelian lattice gauge fields for photons has not been achieved. Here we demonstrate SU(2) lattice gauge fields for photons in the synthetic frequency dimensions31,32, a playground to study lattice physics in a scalable and programmable way. In our lattice model, we theoretically observe that homogeneous non-Abelian lattice gauge potentials induce Dirac cones at time-reversal-invariant momenta in the Brillouin zone. We experimentally confirm the presence of non-Abelian lattice gauge fields by two signatures: linear band crossings at the Dirac cones, and the associated direction reversal of eigenstate trajectories. We further demonstrate a non-Abelian scalar lattice gauge potential that lifts the degeneracies of the Dirac cones. Our results highlight the implications of non-Abelian lattice gauge fields in topological physics, and provide a starting point for demonstrations of emerging non-Abelian physics in the photonic synthetic dimensions. Our results may also benefit photonic technologies by providing controls of photon spins and pseudo-spins in topologically non-trivial ways33.

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Fig. 1: Lattice model with non-Abelian gauge fields.
Fig. 2: Experimental setup to create non-Abelian lattice gauge fields in the photonic synthetic frequency dimension.
Fig. 3: Band structure measurements of the lattice Hamiltonian in the photonic synthetic frequency dimension.
Fig. 4: Tomographic eigenstate reconstructions and observation of eigenstate trajectories.

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Data availability

The data that support the results in this paper are available online at GitHub (https://github.com/dali-cheng/nonabelian).

Code availability

The codes that support the results in this paper are also available online at GitHub (https://github.com/dali-cheng/nonabelian).

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Acknowledgements

This work is supported by MURI projects from the US Air Force Office of Scientific Research (grant nos. FA9550-18-1-0379 and FA9550-22-1-0339). We thank D. A. B. Miller for providing laboratory space and equipment. We acknowledge discussions with H. Wang, Y. Yang and K. Van Gasse and help from L. Skaling with experiments. K.W. acknowledges financial support from the Ministère de l’Économie, de l’Innovation et de l’Énergie of Québec. C.R.-C. is supported by a Stanford Science Fellowship.

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Authors and Affiliations

  1. Edward L. Ginzton Laboratory, Stanford University, Stanford, CA, USA

    Dali Cheng, Charles Roques-Carmes, Eran Lustig, Olivia Y. Long, Heming Wang & Shanhui Fan

  2. Department of Electrical Engineering, Stanford University, Stanford, CA, USA

    Dali Cheng & Shanhui Fan

  3. Department of Physics, McGill University, Montreal, Quebec, Canada

    Kai Wang

  4. Department of Applied Physics, Stanford University, Stanford, CA, USA

    Olivia Y. Long & Shanhui Fan

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  1. Dali Cheng
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Contributions

D.C., K.W. and S.F. conceived the original idea; D.C., K.W. and E.L. designed the experiment; D.C. and C.R.-C. acquired the data with contributions from O.Y.L.; D.C. and C.R.-C. analysed the data; D.C. performed theoretical and numerical studies with contributions from K.W. and H.W.; S.F. supervised the research; D.C., C.R.-C. and S.F. wrote the paper with inputs from all authors; and D.C., C.R.-C. and S.F. revised the paper with inputs from all authors.

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Correspondence to Shanhui Fan.

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Extended data figures and tables

Extended Data Fig. 1 Dirac cone as a signature of non-Abelian lattice gauge fields.

Theoretical band structure of Hamiltonian equation (2) in the vicinity of Γ: (kx, ky) = (0, 0), for (a) non-Abelian gauge potentials Ax = (π/2)σz and Ay = (π/2)σx, and (b) Abelian gauge potentials Ax = Ay = (π/2)σz.

Extended Data Fig. 2 Illustration of loops in the Brillouin zone and topological charges of Dirac points.

The Dirac points at Γ: (0, 0), M: (π, π) and those at X: (π, 0), (0, π) are of opposite charges, as indicated by the ± signs. (a) The black solid line represents the sampling loop Lφ=π/2. (b) An auxiliary loop L* by moving Lφ=π/2 to the +kx direction by π/3 but without touching any Dirac points. The loop hence circles around Dirac points at Γ and M. The eigenstate trajectories have the same handedness for Lφ=π/2 and L*. (c) The black solid line represents the sampling loop Lφ=−π/2. Note that Lφ=−π/2 can be viewed as the remaining part of L* after the circles in the vicinity of Dirac points (as shown in dashed lines) are disconnected. Thus the difference between the eigenstate trajectories along Lφ=±π/2 exists in the topological charges of Dirac points at Γ and M.

Extended Data Fig. 3 A detailed illustration of the experimental setup.

Electronic connections are shown in black thin lines, single-mode fibres in blue lines, polarization-maintaining fibres in purple lines, and free-space collimated beams in orange thick lines. The free-space section is highlighted in the orange box. PC, polarization controller; PSG, polarization state generator; C1–C3, fibre–free-space collimators; BS, beam splitter; FC, fibre coupler; PBS/C, polarizing beam splitter/combiner; EOM, electro-optic modulator; EDFA, erbium-doped fibre amplifier; FPGA, field-programmable gate array; RF amp., radiofrequency amplifier; PD, photodetector.

Extended Data Fig. 4 Comparison between spectra \(\widetilde{{\boldsymbol{\xi }}}({\boldsymbol{\delta }}{\boldsymbol{\omega }},{\boldsymbol{t}})\) and \(\mathop{{\boldsymbol{\xi }}}\limits^{\approx }({\boldsymbol{\delta }}{\boldsymbol{\omega }},{\boldsymbol{t}})\).

The definitions of \(\widetilde{\xi }\) and \(\mathop{\xi }\limits^{\approx }\) can be found in the Methods section. (a) The resonance features in \(\widetilde{\xi }\) near δω = 0. (b) The resonance features in \(\widetilde{\xi }\) near δω = 0.5 ΩR. (c) The quantity \(\mathop{\xi }\limits^{\approx }\) as defined in equation (19) in the Methods section. (a) and (b) are experimentally obtained with φ = 0, g = 0.14 ΩR, and vertically polarized input light. (c) is obtained by averaging (a) and (b), and is a replica of Fig. 3d. The colour scale is normalized to [0, 1] for each individual subplot.

Extended Data Fig. 5 Simulations of band structure measurements with different input polarizations.

We calculate and plot \(\mathop{\xi }\limits^{\approx }(\delta \omega ,t)\) as described in this figure. The input polarizations are set to be H (horizontally polarized), X (+45° linearly polarized), Y (+135° linearly polarized), L (left circularly polarized), and R (right circularly polarized). The results of vertically polarized input are shown in Fig. 3c. We take the modulation strength g = 0.14 ΩR and round trip power loss in the resonator as 1.74 dB. For each value of φ, the colour scale is the same as that in Fig. 3c to facilitate the comparison between conditions where φ is identical but ψin is different.

Extended Data Fig. 6 Equirectangular projections of experimental eigenstate trajectories.

The purpose of this figure is to show an alternative presentation of the experimental data in Fig. 4. (a) Data points with φ = π/2, corresponding to the first row in Fig. 4d. (b) Data points with φ = −π/2, corresponding to the second row in Fig. 4d. In this figure, a data point \((\langle {\sigma }_{x}\rangle ,\langle {\sigma }_{y}\rangle ,\langle {\sigma }_{z}\rangle )\) in Fig. 4d is first rotated to \((\langle {\sigma }_{z}\rangle ,\langle {\sigma }_{x}\rangle ,\langle {\sigma }_{y}\rangle )\) and then projected onto a two-dimensional plane using its azimuthal and polar angles on the Bloch sphere. This is to avoid the drastic changes in the projected trajectories when the data points get close to the north and south poles of the Bloch sphere. The colouring scheme of the data points is consistent with Fig. 4d. The direction reversal of eigenstates manifests in that the eigenstate trajectory overall travels to the left in (a) and to the right in (b). The letters A to E indicate the sequence of the data points along the eigenstate trajectories and are used to guide the eye.

Extended Data Fig. 7 The manifolds \(\boldsymbol{\mathcal{M}}\) in the three-dimensional d-space, and the corresponding band structures, with different values of A0.

When dx ≠ ±dy, that is, the vector gauge potentials are non-Abelian, the manifold \({\mathcal{M}}\) has a geometry of a double-covered rhombus centered at d0. The manifold \({\mathcal{M}}\) is a closed surface: it is composed of two pieces of rhombus-shaped sheets overlapping with each other and “glued” together on the four edges. (a) When d0 = 0, the origin of d-space resides within \({\mathcal{M}}\) and therefore Dirac points exist in the band structure, regardless of the values of Ax and Ay. In this plot we have Ax = (π/2)σz, Ay = (π/2)σx, A0 = 0. (b) When d0 ≠ 0, it is possible that \({\mathcal{M}}\) is translated such that the origin of d-space resides outside \({\mathcal{M}}\), and hence the band structure is gapped. In this plot we have Ax = (π/2)σz, Ay = (π/2)σx, A0 = 0.9σy. In this figure the origins of the d-spaces are highlighted by the black dots.

Extended Data Fig. 8 Band structure measurements with different strengths of non-Abelian scalar potentials.

(a) Theoretical band energies along the sampling line ky = 3kx (mod 2π), calculated by diagonalizing the lattice Hamiltonian. (b), (c) Simulated and measured results of time-dependent transmission spectra of the system. In (b) and (c), we plot the quantity \(\mathop{\xi }\limits^{\approx }(\delta \omega ,t)=\frac{1}{3}[\widetilde{\xi }(\delta \omega ,t)+\widetilde{\xi }(\delta \omega +{\varOmega }_{{\rm{R}}}/3,t)+\widetilde{\xi }(\delta \omega +2{\varOmega }_{{\rm{R}}}/3,t)]\). In the simulation, we use the interleaving modulation scheme, take the modulation strength g = 0.14 ΩR and round trip power loss in the resonator as 0.87 dB. In both the simulation and experiment, the input state is fixed to be left circularly polarized, and the color scale is normalized to [0, 1] for each individual subplot. The time origin in (c) is chosen such that the comparison with (a) and (b) is transparent.

Extended Data Fig. 9 Chiral edge states on a Jackiw–Rebbi-type interface enabled by non-Abelian scalar lattice gauge potentials.

(a) The lattice model under consideration. In the red, top-left region of the lattice, the scalar potential is +A0, and in the blue, bottom-right region the scalar potential is −A0. Here we take Ax = (π/2)σz, Ay = (π/2)σx, and A0 = 0.1σy + 0.75σz. (b) Projected band structure of the lattice model. \({k}_{{\rm{p}}}=({k}_{x}+{k}_{y})/\sqrt{2}\) is the wavevector parallel to the interface along the (11) direction. Chiral edge states are observed in the band gap. (c) Profile of the eigenstate at \({k}_{{\rm{p}}}=0.1{\rm{\pi }}/\sqrt{2}\) and E = 0.13, as indicated by the black star in (b). The left side of the plot represents the normalized amplitudes of the spin-down component of the eigenstate, and the right side represents those of the spin-up component. The interface is placed at lattice site number = 0.

Extended Data Fig. 10 Proposed implementation of the QWZ model involving non-Abelian scalar potentials.

(a) Proposed experimental setup. The up- and down-arrows represent the polarization (pseudo-spin) components supported by the fibre waveguide. Five individual polarization-manipulating branches are represented by five purple boxes. (b) The configuration of each polarization-manipulating branch. The single-mode fibre waveguide is shown in blue, and the polarization-maintaining waveguide shown in purple. In (a) and (b), BS, non-polarizing beam splitter, PC, polarization controller, PBS/C, polarizing beam splitter/combiner, EOM, electro-optic modulator, FC, fibre coupler, PD, photodetector. (c), (d) Theoretical and simulation results of the band structures of the QWZ Hamiltonian, implemented by the experimental configuration in (a). In (c) and (d), we take M = 3, φ = 0, t1 = t2 = t3 = 1, and m0 ∈ {0, 0.5}. In (d), we set the modulation strength g = 0.016ΩR in the weak modulation limit, and the round trip power loss in the resonator as 0.04 dB. We also sum the spectrum signals with orthogonal input polarizations to improve the uniformity of the intensity, that is, the plotted quantity is \(\mathop{\xi }\limits^{\approx }(\delta \omega ,t,{\psi }_{{\rm{in}}}=H)+\mathop{\xi }\limits^{\approx }(\delta \omega ,t,{\psi }_{{\rm{in}}}=V)\). The colour scale is normalized to [0, 1]. (e) Modulation signals τj↑,↓(t) and polarization rotations Uj for implemeting the QWZ model. In the modulation signal, ↑ corresponds to the − sign and ↓ corresponds to the + sign, respectively. In this table τj↑,↓(t) has a period of TR, and the modulation signals are not interleaved.

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Cheng, D., Wang, K., Roques-Carmes, C. et al. Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions. Nature 637, 52–56 (2025). https://doi.org/10.1038/s41586-024-08259-2

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