Direct extraction of topological Zak phase with the synthetic dimension

Topological photonic materials exhibit exotic properties such as protected edge states, unidirectional transport, higher-order corner states, and phenomena enabled by synthetic dimensions, non-equilibrium physics, nonlinearities, non-Hermiticity, and quantum effects, with applications in integrated photonics. One-dimensional (1D) topology is characterized by the Zak phase, the integral of the Berry connection over the Brillouin zone. In the SSH model, the Zak phase is 0 (trivial) or π (non-trivial), corresponding to winding number W = 0 or 1. Photonic demonstrations of probing the Zak phase in SSH-like systems have used schemes such as Bloch oscillations with Ramsey interferometry, mean chiral displacement, leaky lattices, or symmetry breaking in extended SSH models. However, because bulk band dispersions of trivial and non-trivial SSH phases are identical under exchange of intra- and inter-cell couplings, topological information cannot be directly distinguished from bulk bands in standard platforms. Synthetic frequency dimensions, formed by coupling resonant frequency modes via external modulation, offer reconfigurable lattice connectivity and have enabled phenomena such as Hall ladders with synthetic flux, dynamic band structures, non-Hermitian topology with asymmetric spectral hopping, and flat bands. Yet lattices with nonuniform connectivities (e.g., SSH, higher-order models) have largely remained theoretical in frequency synthetic space. This work constructs a synthetic SSH lattice along the frequency axis using two coupled rings with bichromatic modulation, and demonstrates that the Zak phase can be directly extracted from bulk transmission spectra via the time-resolved projected band structure, overcoming the traditional limitation.

Prior approaches to measure Zak phase in photonic systems include Bloch oscillations combined with Ramsey interferometry, mean chiral displacement of wavepackets, use of leaky photonic lattices, and breaking chiral symmetry in extended SSH models. Synthetic frequency dimensions have been used to engineer effective magnetic flux (Hall ladders), dynamic band structures via off-resonant modulation, non-Hermitian topology via asymmetric spectral hoppings, and flat bands via synthetic stub lattices, typically with uniform mode coupling. Unequal couplings enabling nonuniform connectivity (e.g., SSH, quadrupole higher-order insulators) have been mainly proposed theoretically in the frequency dimension, motivating experimental realization.

Theory and model: Two identical coupled ring resonators (rings A and B) support equally spaced resonant modes An and Bn at frequencies ωn = ω0 + nΩ (Ω is the FSR). Evanescent coupling between rings with strength κ splits degenerate modes into symmetric (Cn at ωn + κ) and antisymmetric (Dn at ωn − κ) supermodes, producing alternating frequency spacings Ω1 = 2κ and Ω2 = Ω − 2κ. An electro-optic phase modulator (EOM) inside ring A applies bichromatic modulation J(t) = 4g1 cos(Ω1 t + φ1) + 4g2 cos(Ω2 t + φ2). Starting from the coupled-ring Hamiltonian, transforming to the symmetric/antisymmetric basis and using the rotating-wave approximation yields a tight-binding model H ≈ Σn[(g1 e^{iφ} c†n dn + g2 e^{-iφ} d†n cn+1) + h.c.], which maps to a 1D SSH lattice along the synthetic frequency dimension with intra-cell and inter-cell hoppings g1 and g2 between sites Dn and Cn. Defining the unit cell as (Dn, Cn) fixes the topology (opposite choice flips it). In momentum ky (reciprocal to frequency index), the Bloch Hamiltonian has off-diagonal G(ky) = g1 e^{-iφ} + g2 e^{i(ky+φ)}, eigenvalues Ek,m = m sqrt(g1^2 + g2^2 + 2 g1 g2 cos(ky/2 + φ1 + φ2)) with m = ±1, and eigenstates |ψk,m⟩ with relative phase arg(G). The Zak phase φZak = (1/2π) ∫_{-π}^{π} ∂ky arg(G) dk distinguishes phases: φZak = π for g1 < g2 (non-trivial), φZak = 0 for g1 ≥ g2 (trivial). Although exchanging g1 and g2 leaves band shapes invariant, it changes the winding number and Zak phase. Experimental platform: Two 10.2 m fiber rings (A and B) are coupled by a 50:50 2×2 fiber coupler. The measured FSR is Ω/2π ≈ 20 MHz. Without modulation, supermode splitting is ≈ 2π×6.67 MHz (thus Ω1 = 2π×6.67 MHz and Ω2 = 2π×13.33 MHz). Ring A contains an EOM driven by an RF signal V1 cos(Ω1 t) + V2 cos(Ω2 t) to realize staggered coupling strengths (g1 ∝ V1, g2 ∝ V2). Components include polarization controllers, SOAs, EDFA, DWDM, photodiode, AWG, and oscilloscope. Time-resolved projected band-structure spectroscopy: Transmission spectra at the drop port are measured by scanning the input laser frequency near ω0 ± κ, yielding output fields Sout^{±κ}(t) whose resonances occur at detunings matching the eigenvalues εk,m. Due to discrete translation symmetry, ky maps to time t. For fixed detuning, the normalized transmission shows peaks at Δω^{±κ} = εk,m. The spectra are segmented into time slices with window 2π/Ω1 and stacked versus detuning to reconstruct the time-resolved projected band structure. Two groups of asymmetric projected bands (each with two subbands corresponding to m = ±1) appear, separated by 2κ, reflecting projections onto superpositions of Ck and Dk. The envelope follows Ek,m. Under V1 < V2 (e.g., V1 = 1 V, V2 = 3 V), dispersive projected bands are observed; setting V1 = 0, V2 = 3 V (g2/g1 → ∞) yields flat projected bands. Simulations using the derived expressions reproduce the observed bands for g1 = 0.02 Ω and g2 = 0.06 Ω. Zak phase extraction (resonant method): Using simplified expressions for the output field that include the eigenstate phase φ(k), the phase information is encoded in the time-resolved spectra. Selecting a specific band (e.g., m = 1) and resonantly tracking Δω = Ek,m at each ky maximizes its contribution. An intensity parameter S^2 ∝ [1 + m cos(2κ t)] is obtained from the peak of each vertical time slice and normalized to [−1,1] to decode arg(G) and evaluate the Zak phase from the winding of φ(k) across the Brillouin zone. Applying this to bands indicated in the measured spectra yields quantitative φZak values for different coupling configurations.

• Constructed a synthetic SSH lattice along the frequency dimension using two coupled fiber rings with bichromatic modulation that alternates coupling strengths between symmetric and antisymmetric supermodes.
• Demonstrated that, while the SSH bulk band shapes are invariant under g1 ↔ g2 (thus traditionally hiding topology), the time-resolved projected band structures measured via transmission spectroscopy encode the eigenstate phase and reveal topology.
• Experimentally observed two groups of projected bands separated by ~2κ, each with two subbands; for V1 = 1 V, V2 = 3 V (g1 < g2), bands are dispersive; for V1 = 0 V, V2 = 3 V (g2/g1 → ∞), all four bands become flat. Simulations with g1 = 0.02 Ω and g2 = 0.06 Ω matched experiments.
• Using the resonant method on selected bands, extracted Zak phase values corresponding to the two phases: approximately 0 for the trivial configuration and ~0.98π for the non-trivial configuration, directly from bulk spectral data (no edge-state measurements).
• Implemented at telecom wavelengths with ring FSR ≈ 20 MHz and supermode splitting ≈ 6.67 MHz, showcasing a practical, fiber-based platform.

The work addresses the central challenge that bulk band dispersions of the SSH model do not distinguish trivial and non-trivial phases. By measuring time-resolved projected band structures, the method accesses the eigenstate phase arg(G), whose winding over the Brillouin zone determines the Zak phase. This allows direct extraction of topological invariants from bulk responses without relying on edge states or interferometric protocols. The agreement between measured and simulated projected bands validates the theoretical mapping and the spectroscopy technique. The approach is general to other 1D topological models and, in principle, higher-dimensional synthetic lattices, enabling flexible reconfiguration and on-chip-compatible implementations. The contrasting projected spectra across the topological transition may also be leveraged for optical communication functionalities where spectral signatures encode robust information.

This study experimentally realizes a synthetic SSH lattice in the frequency dimension using bichromatically modulated coupled fiber rings and introduces a spectroscopy-based resonant method to directly extract the Zak phase from bulk projected band structures. The approach overcomes the traditional indistinguishability of SSH phases from bulk dispersions by exploiting the eigenstate phase encoded in time-resolved transmission. Experiments retrieve Zak phases near 0 (trivial) and ~0.98π (non-trivial), with results corroborated by theory. The platform is reconfigurable and compatible with telecom technologies, offering a simple route to study and harness topological phases in synthetic dimensions. Future work can extend to higher-dimensional topological invariants, more complex nonuniform connectivities (e.g., higher-order topological phases), and integration with non-Hermitian, nonlinear, or quantum regimes.

• The synthetic lattice is effectively infinite in frequency without hard boundaries; topology depends on defining the unit cell (choice of intra- vs inter-cell coupling), which must be fixed for unambiguous interpretation.
• Analyses often neglect group-velocity dispersion; in practice, gradual FSR variation and off-resonant coupling due to dispersion can introduce deviations from the ideal lattice and limit lattice size.
• Measurements rely on resolving time-resolved projected bands with adequate signal-to-noise; losses (γ), component bandwidths, and modulation linearity constrain spectral resolution and accuracy of phase extraction.
• Reported experiments focus on specific coupling regimes (e.g., V1 < V2); comprehensive mapping over broader parameter spaces and robustness to disorder were not detailed in the provided text.
• Exact publication provides limited affiliation details for all authors; some experimental component specifics (e.g., exact calibration of g1, g2 vs applied voltages) are summarized rather than exhaustively characterized here.