Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions

The authors demonstrate SU(2) lattice gauge fields for photons in the platform of synthetic frequency dimensions.

“Here, we demonstrate non-Abelian lattice gauge fields for photons.”

• Also stated in: § Introduction · boosted

  “Here, we demonstrate SU(2) lattice gauge fields for photons in the platform of synthetic frequency dimensions [42,43], an ideal playground to study lattice physics in a scalable and programmable way.”

• Also stated in: § Discussion · plain

  “We experimentally observe non-Abelian lattice gauge fields for photons using synthetic frequency dimensions created in a photonic resonator system.”

• cites [42] Li, Y., Chan, C. T. & Mazur, E. Dirac-like cone-based electromagnetic zero-index metamaterials. Light Sci. Appl. 10, 203 (2021). sourceCould not verify

  The Introduction discusses Dirac-like cones and zero-index metamaterials, not a demonstration of photon lattice gauge fields. Searches for SU(2), non-Abelian gauge fields, and synthetic frequency dimensions found no passage in our copy addressing the attributed claim. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [43] Yuan, L., Xiao, M., Lin, Q. & Fan, S. Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation. Phys. Rev. B 97, 104105 (2018). sourcePartly supported

  The Abstract (p. 1) says modulation phases create a photonic gauge potential in synthetic space, and Section III (p. 6) proposes a Haldane model using three resonators. Neither passage supports the more specific attribution of SU(2), non-Abelian lattice gauge fields. (LLM, conf=0.88)

  Experimental — accuracy not yet measured

In many existing works Dirac points are observed only at time-reversal-breaking wavevectors, such as the K- and K'-points in a honeycomb lattice.

“However, in many existing works Dirac points are observed only at time-reversal-breaking wavevectors, such as the Kand K'-points in a honeycomb lattice [46][47][48].”

• cites [46] Cheng, D., Lustig, E., Wang, K. & Fan, S. Multi-dimensional band structure spectroscopy in the synthetic frequency dimension. Light Sci. Appl. 12, 158 (2023). sourceCould not verify

  The Results section on two-dimensional non-Hermitian band structures discusses point-gap topology and time-reversal-invariant loops, not Dirac points. Searches for Dirac points, honeycomb lattices, and K/K′ wavevectors found no passage in our copy addressing where Dirac points have been observed in existing works. (LLM, conf=0.75)

  Experimental — accuracy not yet measured
• cites [47] Qi, X.-L., Wu, Y.-S. & Zhang, S.-C. Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors. Phys. Rev. B 74, 085308 (2006). sourceCould not verify

  Section II (p. 1) mentions Haldane’s honeycomb-lattice model, but does not locate its Dirac points. Searches for “Dirac point,” “K points,” and time-reversal-related wavevectors found no passage supporting the claim that many existing works observe Dirac points only at K and K′ or other such wavevectors. (LLM, conf=0.74)

  Experimental — accuracy not yet measured
• cites [48] Dutt, A. et al. Creating boundaries along a synthetic frequency dimension. Nat. Commun. 13, 3377 (2022). sourceCould not verify

  The Results section on the quantum Hall ladder discusses time-reversal symmetry breaking and chiral modes, but not Dirac points or where they occur. Searches for “Dirac,” “honeycomb,” and K/K′ wavevectors found no passage addressing the claimed pattern across existing works. Page numbers are unavailable in our copy. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

If such a closed loop in the Brillouin zone encircles a Dirac point, the winding of the trajectory can be associated with the topological charge of the Dirac point.

“If such a closed loop in the Brillouin zone encircles a Dirac point, the winding of the trajectory can be associated with the topological charge of the Dirac point [44,45].”

• cites [44] Wang, K. et al. Multidimensional synthetic chiral-tube lattices via nonlinear frequency conversion. Light Sci. Appl. 9, 132 (2020). sourceCould not verify

  The Introduction discusses gauge-field flux around an area, and the Results section on quantum walks analyzes reciprocal-space dispersion, but neither relates a trajectory’s winding to a Dirac point’s topological charge. Searches for “Dirac” and “winding number Brillouin zone topological charge” found no passage addressing that relationship. The indexed passages provide section labels but no page numbers. (LLM, conf=0.72)

  Experimental — accuracy not yet measured
• cites [45] Dutt, A. et al. Experimental band structure spectroscopy along a synthetic dimension. Nat. Commun. 10, 3122 (2019). sourceCould not verify

  The Results > Theory section describes Bloch quasimomentum and Floquet band-structure measurements, while Results > Experimental results discusses a Brillouin zone and synthetic gauge potentials; page numbers are unavailable in our copy. Searches for Dirac points, winding, closed loops, and topological charge found no passage connecting a loop around a Dirac point to its charge. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

The resonance locations in the time-dependent transmission spectra correspond to the band energies along the sampling line Lφ: ky = M kx + φ (mod 2π) in the two-dimensional Brillouin zone.

“the resonance locations in the time-dependent transmission spectra correspond to the band energies along the sampling line L φ : k y = M k x + φ (mod 2π) in the two-dimensional Brillouin zone.”

• cites [55] Pang, Z., Wong, B. T. T., Hu, J. & Yang, Y. Synthetic non-Abelian gauge fields for non-Hermitian systems. Phys. Rev. Lett. 132, 043804 (2024). sourceCould not verify

  The paper’s discussion of its Bloch Hamiltonian and eigen-energies (Eq. 3–5; page unavailable) concerns a single momentum k. Searches for transmission spectra and the proposed sampling line found no passage connecting resonance locations to band energies along ky = M kx + φ in a two-dimensional Brillouin zone. (LLM, conf=0.75)

  Experimental — accuracy not yet measured

The one-dimensional lattice with long-range couplings in Fig. 2(a) is equivalent to a two-dimensional lattice with twisted boundary conditions.

“The onedimensional lattice with long-range couplings in Fig. 2(a) is equivalent to a two-dimensional lattice with twisted boundary conditions as shown in Fig. 2(b) [52,53].”

• cites [52] Yang, Y., Zhen, B., Joannopoulos, J. D. & Soljačić, M. Non-Abelian generalizations of the Hofstadter model: Spin–orbit-coupled butterfly pairs. Light Sci. Appl. 9, 177 (2020). sourceCould not verify

  The Introduction describes a two-dimensional Hofstadter lattice, and Materials and methods describes square-lattice models with nearest-neighbor coupling; neither addresses the claimed equivalence. Searches for one-dimensional long-range couplings and twisted boundary conditions found no passage establishing that mapping in our copy. No page numbers were available. (LLM, conf=0.74)

  Experimental — accuracy not yet measured
• cites [53] Liu, V., Yang, Y., Joannopoulos, J. D. & Soljačić, M. Three-dimensional non-Abelian generalizations of the Hofstadter model: Spin–orbit-coupled butterfly trios. Phys. Rev. B 104, 115127 (2021). sourceCould not verify

  The Introduction describes a three-dimensional cubic-lattice model whose Cartesian surfaces reduce to two-dimensional models, not an equivalence between a one-dimensional long-range lattice and a two-dimensional lattice. The source paper’s Fig. 2 caption describes band structures and inversion eigenvalues; searches for long-range couplings and twisted boundary conditions found no passage addressing the claimed equivalence. Page numbers were unavailable in our copy. (LLM, conf=0.74)

  Experimental — accuracy not yet measured

Homogeneous non-Abelian lattice gauge potentials give rise to Dirac cones at time-reversal-invariant momenta in the Brillouin zone.

“Key to our experiment is the theoretical observation that homogeneous non-Abelian lattice gauge potentials give rise to Dirac cones at time-reversalinvariant momenta in the Brillouin zone.”

• Also stated in: § Signatures of non-Abelian lattice gauge fields · boosted

  “We prove that in the lattice model Eq. ( 2), the Dirac cones at time-reversal-invariant momenta occur if and only if the gauge potentials are non-Abelian.”

The experiments confirm the presence of non-Abelian lattice gauge fields by the reversal of the directions of eigenstate trajectories along a loop in the Brillouin zone as the loop passes through the Dirac points.

“Our experiments confirm the presence of non-Abelian lattice gauge fields by two experimental signatures: (1) the presence of such Dirac cones, and (2) the reversal of the directions of eigenstate trajectories along a loop in the Brillouin zone as the loop passes through the Dirac points.”

The experiments confirm the presence of non-Abelian lattice gauge fields by the presence of Dirac cones at time-reversal-invariant momenta.

“Our experiments confirm the presence of non-Abelian lattice gauge fields by two experimental signatures: (1) the presence of such Dirac cones, and (2) the reversal of the directions of eigenstate trajectories along a loop in the Brillouin zone as the loop passes through the Dirac points.”

When a wavevector loop passes the Dirac points, the handedness of the corresponding eigenstate trajectory reverses.

“As detailed in the Methods section, when a wavevector loop passes the Dirac points, the handedness of the corresponding eigenstate trajectory reverses.”

In the model of Eqs. (1) and (2), Dirac points at time-reversal-invariant momenta occur if and only if [A_x, A_y] ≠ 0.

“In our model of Eqs.(1) and ( 2), the Dirac points at time-reversal-invariant momenta occur if and only if [A x , A y ] ̸ = 0 (with proof provided in the Methods section).”

The lattice system of Eq. (1) exhibits observable physical effects even though the non-Abelian gauge potentials are also uniform.

“In contrast, the lattice system of Eq. ( 1) exhibits observable physical effects even though the non-Abelian gauge potentials are also uniform.”

Emergence of Dirac points at time-reversal-invariant momenta is equivalent to d_x ≠ ±d_y.

“Emergence of Dirac points at time-reversal-invariant momenta ⇔ d x ̸ = ±d y .”

The Hamiltonian Eq. (4) has the chiral symmetry σ_y Ĥ(k_x, k_y)σ_y^{-1} = -Ĥ(k_x, k_y).

“The Hamiltonian Eq. ( 4) has the chiral symmetry σ y Ĥ(k x , k y )σ -1 y = -Ĥ(k x , k y ).”

The system realizes a two-dimensional Yang-Mills Hamiltonian.

“Our system realizes a two-dimensional Yang-Mills Hamiltonian.”

The signature that the eigenstate trajectories on the Bloch sphere have reversed directions for φ = π/2 and φ = -π/2 persists outside the weak modulation approximation.

“The signature, that the eigenstate trajectories on the Bloch sphere have reversed directions for φ = π/2 and φ = -π/2, persists outside the weak modulation approximation.”

The authors experimentally observe non-Abelian lattice gauge fields for photons using synthetic frequency dimensions created in a photonic resonator system.

“We experimentally observe non-Abelian lattice gauge fields for photons using synthetic frequency dimensions created in a photonic resonator system.”

The theoretical band energies along the sampling line show linear band crossings at both Γ and M points for φ ∈ {0, π}.

“The corresponding theoretical band energies along the sampling line are shown in the top and bottom pan- els of Fig. 3(b), where the linear band crossings are evident at both Γ and M points.”

In the weak modulation limit g ≪ Ω_R, the evolution dynamics of the photon state in the resonator is governed by a Schrödinger-like equation, with a Hamiltonian corresponding to Eq. (1).

“In the weak modulation limit g ≪ Ω R , the evolution dynamics of the photon state in the resonator is governed by a Schrödinger-like equation, with a Hamiltonian corresponding to Eq. ( 1) (see the Methods section for details of the derivation).”

• AssumptionThe authors state this as an assumption: The weak modulation limit g ≪ Ω_R applies for the derivation mapping the resonator dynamics to the lattice Hamiltonian.

The method developed in this paper allows the first fully tomographic reconstruction of eigenstates in a synthetic frequency lattice.

“We emphasize that the method we developed in this paper allows us to perform the first fully tomographic reconstruction of eigenstates in a synthetic frequency lattice.”

Experimental results of the eigenstate trajectories along loops L_{φ=±π/2} show the expected reversal of winding directions.

“The last column in Fig. 4(b) shows experimental results of the eigenstate trajectories along loops L φ=±π/2 , and we observe the expected reversal of winding directions.”

In Hamiltonian Eq. (4), the Dirac points at Γ, M and those at X are of opposite charges.

“In Hamiltonian Eq. ( 4), the Dirac points at Γ, M and those at X are of opposite charges.”

The sampling line L_φ: k_y = M k_x + φ (mod 2π) forms a non-contractible loop that cannot be characterized by an 'interior' and an 'exterior'.

“The sampling line of our interest, L φ : k y = M k x + φ (mod 2π), forms a non-contractible loop that cannot be characterized by an “interior” and an “exterior.””

The Hamiltonian associated with Eq. (15) is described by Eq. (1), up to a constant factor, with non-Abelian gauge potentials A_x = (π/2)σ_z and A_y = (π/2)σ_x.

“The Hamiltonian associated with Eq. ( 15) is described by Eq. ( 1), up to a constant factor, with non-Abelian gauge potentials A x = (π/2)σ z and A y = (π/2)σ x .”

• AssumptionThe authors state this as an assumption: The weak modulation limit g ≪ Ω_R applies for the derivation mapping the resonator dynamics to the lattice Hamiltonian.

The steady-state polarization in the resonator at a given wavevector k, represented by a Jones vector ψ_ss, is related to the input polarization state ψ_in by the Green's function of the system.

“The steadystate polarization in the resonator at a given wavevector k, represented by a Jones vector ψ ss , is related to the input polarization state ψ in by the Green's function of the system:”

In the lattice model Eq. (2), the Dirac cones at time-reversal-invariant momenta occur if and only if the gauge potentials are non-Abelian.

“We prove that in the lattice model Eq. ( 2), the Dirac cones at time-reversal-invariant momenta occur if and only if the gauge potentials are non-Abelian.”

Eq. (15) is subject to the twisted boundary condition that connects lattice sites (0, y) and (M - 1, y - 1).

“Eq. ( 15) is subject to the twisted boundary condition that connects lattice sites (0, y) and (M -1, y-1): ψ -1,y = ψ M -1,y-1 e -iφ , ψ M,y-1 = ψ 0,y e +iφ .”

The experimental measurements display resonant features in the spectra that closely match the theoretical band structures and simulations.

“And our experimental measurements display resonant features in the spectra that closely match the theoretical band structures and simulations.”

[A_x, A_y] ≠ 0 if and only if d_x ≠ ±d_y.

“[A x , A y ] ̸ = 0 ⇔ d x ̸ = ±d y ;”

In experimentally obtained ξ, there are energy levels near both δω = 0 and δω = 0.5ΩR.

“In ξ, there are energy levels near both δω = 0 and δω = 0.5Ω R .”

The signal intensities at resonances are not uniform throughout the Brillouin zone.

“As shown in Figs.3(c, d), the signal intensities at resonances are not uniform throughout the Brillouin zone.”

Using tomographic reconstruction methods, the reversal of the directions of eigenstate trajectories as the sampling lines move across Dirac points in the Brillouin zone is observed.

“Using such methods, we are able to observe the reversal of the directions of eigenstate trajectories as the sampling lines move across Dirac points in the Brillouin zone.”

The band structure measurement results depend on the choice of input polarization state ψin.

“Finally, we notice that the band structure measurement results depend on the choice of input polarization state ψ in .”

The lattice Hamiltonian of Eq. (1) exhibits non-Abelian gauge field physics originating from the lack of commutativity of gauge potentials.

“Therefore, the lattice Hamiltonian of Eq. ( 1) exhibits non-Abelian gauge field physics originating from the lack of commutativity of gauge potentials.”

For a photonic two-band model, having a Dirac point at time-reversal-invariant momenta, such as at the Γ point, is unusual as it requires time-reversal-symmetry breaking.

“For a photonic two-band model, having a Dirac point at timereversal-invariant momenta, such as at the Γ point of k = (0, 0), is unusual as it requires time-reversal-symmetry breaking.”

• AssumptionRests on an assumption: The pseudo-spin of a photon is not changed under the time-reversal operation.

The emergence of Dirac points with linear band crossings is a unique signature of the non-Abelian lattice gauge fields in the model.

“The emergence of such Dirac points with linear band crossings is therefore a unique signature of the non-Abelian lattice gauge fields in our model.”

The lattice Hamiltonian of Eq. (1) describes path-dependent spin rotations as a particle travels between lattice sites.

“The Hamiltonian of Eq. ( 1) describes path-dependent spinrotations as a particle travels between lattice sites.”

These results further confirm the emergence of non-Abelian lattice gauge fields for photons in the synthetic frequency dimension.

“These results further confirm the emergence of non-Abelian lattice gauge fields for photons in the synthetic frequency dimension.”

The experimental setup can be modelled by the lattice Hamiltonian Eq. (1) under appropriate assumptions.

“Our experimental setup can be modelled by the lattice Hamiltonian Eq. ( 1) under approapriate assumptions.”

• AssumptionThe authors state this as an assumption: The weak modulation limit g ≪ Ω_R applies for the derivation mapping the resonator dynamics to the lattice Hamiltonian.

This corresponds to an absolute value of topological charge of 2, the sum of the charges of the Dirac points at Γ and at M.

“This corresponds to an absolute value of topological charge of 2, the sum of the charges of the Dirac points at Γ and at M .”

By extracting the on-resonance intensities in the time-dependent transmission spectra for different input polarizations, the Stokes parameters of the eigenstate can be reconstructed.

“Therefore, by extracting the onresonance intensities in the time-dependent transmission spectra ξ(δω, k, ψ in ) for different input polarizations ψ in , we can reconstruct the Stokes parameters of the eigenstate.”

The signatures arising from the non-Abelian nature of the gauge potentials in the lattice system cannot be achieved in an Abelian system.

“The theoretical results above thus highlight the unique signatures that arise from the non-Abelian nature of the gauge potentials in the lattice system: these signatures cannot be achieved in an Abelian system.”

The non-uniformity of the signal strength at resonance originates from the variation in the overlap between the eigenstate and the fixed input polarization.

“The non-uniformity of the signal strength at resonance originates from the variation in the overlap between the eigenstate and the fixed input polarization.”

When traced along a closed loop in the Brillouin zone, an eigenstate traces out a trajectory that winds around the origin of the Bloch sphere for an integer number of times.

“When traced along a closed loop in the Brillouin zone, an eigenstate thus traces out a trajectory that winds around the origin of the Bloch sphere for an integer number of times.”

These physical effects arise because the non-commutativity of the non-Abelian gauge potentials is gauge-invariant: it cannot be removed by any gauge transformation.

“These physical effects arise because the non-commutativity of the non-Abelian gauge potentials is gauge-invariant: it cannot be removed by any gauge transformation.”

In this case, the Dirac point is manifested by the change in the eigenstate trajectory as the sampling line moves across the Dirac point.

“In this case, the Dirac point is manifested by the change in the eigenstate trajectory as the sampling line moves across the Dirac point, as illustrated in Fig. 6.”

These results indicate the robustness of the non-Abelian lattice gauge fields outside the weak modulation regime.

“These results indicate the robustness of the non-Abelian lattice gauge fields outside the weak modulation regime.”

The linear band crossings at kx ∈ {0, π} revealed by the band structure measurements are one of the signatures of the non-Abelian lattice gauge fields.

“The linear band crossings at k x ∈ {0, π} revealed by the band structure measurements are one of the signatures of the non-Abelian lattice gauge fields.”

A lattice system with a uniform Abelian gauge potential does not have any physically observable effects associated with the gauge potential, since the potential can always be made to vanish through gauge transformations.

“A lattice system with a uniform Abelian gauge potential does not have any physically observ-able effects associated with the gauge potential, since the potential can always be made to vanish through gauge transformations.”

The poles of K−1(δω, t) are associated with the resonances that match the band energies of the lattice Hamiltonian.

“Here K -1 (δω, t) is the Green's function of the system, and the poles of K -1 (δω, t) are associated with the resonances that match the band energies of the lattice Hamiltonian.”

Therefore the eigenstates are pinned onto a circle defined as the intersection of the ⟨σ_x⟩-⟨σ_z⟩ plane with the Bloch sphere.

“Therefore the eigenstates are pinned onto a circle defined as the intersection of the ⟨σ x ⟩-⟨σ z ⟩ plane with the Bloch sphere.”

The linear band crossings at Γ, X, and M points are unique signatures of the non-Abelian lattice gauge fields.

“Therefore, the linear band crossings at Γ, X, and M points are unique signatures of the non-Abelian lattice gauge fields.”

The information encoded in ≈ξ involves contributions from input polarization states ψin and Y(δω, t)ψin.

“The information encoded in ≈ ξ involves contributions from input polarization states ψ in and Y (δω, t)ψ in .”

The intensity signal ξ that one measures is the projection of the eigenstate onto the input state.

“The intensity signal ξ that one measures is the projection of the eigenstate onto the input state.”

The interference terms in the last line of Eq. (19) are artifacts generated by the interleaving modulation scheme.

“The interference terms in the last line of Eq. ( 19) are artifacts generated by the interleaving modulation scheme.”

The steady state is a linear superposition of eigenstates.

“such that the steady state is a linear superposition of eigenstates:”

The existence of energy levels near δω = 0.5ΩR is due to the use of the interleaving modulation scheme as described in Eq. (10), where the modulation signal has a fundamental temporal period of 2TR.

“The existence of energy levels near δω = 0.5Ω R is due to the use of the interleaving modulation scheme as described in Eq. ( 10), where the modulation signal has a fundamental temporal period of 2T R .”

The non-uniform signal intensities at resonances arise due to the wavevector dependency in the overlap between the eigenstate of the lattice Hamiltonian and the input polarization state.

“This arises due to the wavevector dependency in the overlap between the eigenstate of the lattice Hamiltonian and the input polarization state -which will be further discussed in the next section.”

The reversal of the directions of eigenstate trajectories as sampling lines move across Dirac points is another signature of the non-Abelian lattice gauge fields.

“This is another signature of the non-Abelian lattice gauge fields.”

The steady-state polarization ψss(δω, k, ψin) in the resonator is related to the input polarization ψin by the Green's function 1/(δω − Ĥ(k) + iγ0/2).

“The steady-state polarization ψ ss (δω, k, ψ in ) in the resonator is related to the input polarization ψ in by the Green's function: ψ ss (δω, k, ψ in ) = 1 δω -Ĥ(k) + iγ 0 /2 ψ in .(21)”

The use of Eq. (20) allows determination of the location of the resonances of the system without the artifacts of the interference terms in Eq. (19).

“The use of Eq. ( 20) allows us to determine the location of the resonances of the system without the artifacts of the interference terms in Eq. ( 19).”

The final spin state is dependent on the path because the spin rotations associated with the lattice links are not commutative.

“The final spin state is dependent on the path because the spin rotations associated with the lattice links are not commutative.”