Introduction

Last few decades have witnessed rapid advances of topological photonic materials with exotic properties, such as topologically protected edge states, unidirectional light transport, high-order topological corner states, topological defects, novel topological phases and phenomena produced in combination with synthetic dimensions, non-equilibrium physics, nonlinearities, non-Hermiticity and quantum effects, which hold important applications in integrated photonic devices1,2,3,4,5,6,7,8,9,10,11,12,13. The topological phases of matter can be classified by their topological invariants14,15,16,17. For example, the topology of a one-dimensional (1D) system is characterized by the Zak phase, obtained by integrating the Berry curvature over the first Brillouin zone18,19. In the well-known Su-Schriefffer-Heeger (SSH) model20, the Zak phase can take two values, which are 0 for the topologically trivial case and π for the topologically non-trivial case, corresponding to the winding numbers of \({{{\mathcal{W}}}} = 0\) and \({{{\mathcal{W}}}} = 1\), respectively21,22. Probing the Zak phase in 1D photonic systems including the photonic SSH model23,24,25,26,27,28,29, has been widely demonstrated within several experimental schemes, such as combining Bloch oscillations and Ramsey interferometry24, implementing the mean chiral displacement of a particle’s wavepacket25, using leaky photonic lattices26, and breaking the chiral symmetry in extended SSH models27,28. However, due to identical shapes of the bulk band structure in both trivial and non-trivial cases for the SSH lattice, the topological information such as the Zak phase cannot be directly distinguished from the bulk band structure in the current platforms30.

Synthetic frequency dimension constructed by coupling the frequency degree of freedom of light has manifested as a powerful platform for creating lattices with artificial connectivities and achieving unusual functionalities that are hard to be achieved in real space31,32,33,34,35,36,37,38. Interesting physics associated with the synthetic frequency dimension have been reported by engineering the connectivity through external modulations, such as the Hall ladder with the effective magnetic flux39, the dynamic band structure with the off-resonant modulation40, non-Hermitian topology with asymmetric spectral hoppings41,42, and the flat band with the synthetic stub lattice43, where equally-spaced frequency modes are coupled with uniform modulations. However, lattice structures formed by nonuniform connectivities between sites hold richer physics in real space44,45,46,47,48,49, which are only proposed in theory in synthetic space with the frequency dimension, including non-Hermitian SSH lattices50 and quadrupole higher-order topological insulators51. Therefore, experimental implementation of such unequal couplings between lattice sites is crucial to bring those theoretical proposals into practice, which may also greatly promote the development of synthetic frequency dimension towards constructing more complex lattice structures beyond geometric dimensionality44,45,46,47,48,49.

In this work, we demonstrate the extraction of Zak phase from the bulk band structure of the 1D synthetic SSH model constructed along the frequency dimension of two coupled ring resonators. The symmetric and antisymmetric supermodes in the ring resonator system are connected by the electro-optic phase modulator (EOM), which provides bichromatic sinusoidal modulations at different amplitudes. Such configuration can connect multiple photonic molecules52 and then form a 1D SSH model along the frequency axis of light, where the topology is characterized by the Zak phase (0 or π). We show that our system possesses its unique feature that the identical shapes of the corresponding band structures under different topological cases can be broken, due to distinct projections of the band structures onto superpositions of the two supermodes. We show that the topological phase information is naturally encoded in the time-resolved projected band structure, which can be extracted from the transmission spectra by choosing the input frequency resonant with the eigenvalue in the momentum space reciprocal to the frequency dimension. Such theoretical proposal is then validated in experiments performed at the telecom wavelength, where measurements in the non-trivial and trivial phases are performed by flexibly reversing the modulation strengths in the bichromatic signal to obtain spectral transmissions. Zak phase values (~ 0 and 0.98π) are then extracted in different topological phases. Our scheme to extract Zak phase from the bulk band structure projected onto supermodes holding the phase information is fundamentally different from other systems holding the SSH model24,25,26,27,28, and is universal to other topological models53,54,55,56,57. Therefore, our work points out a simple route in exploring topological phases of matter with experimental feasibility and reconfigurability in the synthetic frequency dimension, and also holds potential applications in optical communications.

Results

Construction of synthetic SSH model

To construct the equivalent SSH model in the synthetic frequency dimension, we start with considering two identical ring resonators labeled as ring A and ring B in Fig. 1(a). In the absence of group-velocity dispersion, ring A (B) supports equally-spaced resonant modes defined as An (Bn) at frequency ωn = ω0 + nΩ [see Fig. 1(b)], where ω0 is the central frequency, n is the index of the nth mode, and Ω is the free spectral range (FSR) for ring A (B). Modes An and Bn at the same n can be coupled by the evanescent wave or the fiber coupler between the two rings, with coupling strength κ. It then leads to mode splitting with the hybridization of the resonant modes at frequency ωn into the symmetric supermode Cn at the frequency ωn + κ and the antisymmetric supermode Dn at the frequency ωn − κ52,58, which thus constructs unequally spaced synthetic sites in the frequency dimension alternatively separated by Ω1 ≡ 2κ and Ω2 ≡ Ω − 2κ as illustrated in Fig. 1(c). An EOM is placed only inside ring A with the external bichromatic signal

$$J\left( t \right) = 4g_1\cos \left( {{\Omega}_1t + \phi _1} \right) + 4g_2\cos \left( {{\Omega}_2t + \phi _2} \right)$$
(1)

where 4g1, 4g2 and ϕ1, ϕ2 are the modulation amplitudes and phases. One then obtains the corresponding Hamiltonian of the system in Fig. 1(a) as

$$H\left( t \right) = \mathop {\sum}\limits_n {\omega _n\left( {a_n^{\dagger} a_n + b_n^{\dagger} b_n} \right)} + \kappa \mathop {\sum}\limits_n {\left( {a_n^{\dagger} b_n + b_n^{\dagger} a_n} \right)} + \mathop {\sum}\limits_{n,n\prime } {J\left( t \right)a_n^{\dagger} a_{n\prime }}$$
(2)

where an and bn (\(a_n^{\dagger}\) and \(b_n^{\dagger}\)) are the annihilation (creation) operators for the modes An and Bn, respectively. One notes that the last term of Eq. (2) indicates that the applied modulation is off-resonant with resonant modes in ring A. By replacing an and bn with operators \(c_n = \left( {a_n + b_n} \right)/\sqrt 2\) and \(d_n = \left( {a_n - b_n} \right)/\sqrt 2\), and taking the rotating-wave approximation, one can rewrite the Hamiltonian of Eq. (2) into

$$\begin{array}{l}{H_{{{{\mathrm{RWA}}}}} = \mathop {\sum}\limits_n {\left( {\omega _n + \kappa } \right)} c_n^{\dagger} c_n + \mathop {\sum}\limits_n {\left( {\omega _n - \kappa } \right)} d_n^{\dagger} d_n}\\\qquad\quad\;\;\; { + \mathop {\sum}\limits_n {\left[ {g_1e^{ - i\left( {2\kappa t + \phi _1} \right)}c_n^{\dagger} d_n + g_2e^{ - i\left[ {\left( {{\Omega} - 2\kappa } \right)t + \phi _2} \right]}d_n^{\dagger} c_{n - 1} + h.c.} \right]} }\end{array}$$
(3)
Fig. 1: Configuration of the 1D synthetic SSH model.
figure 1

a Illustration of two identical coupled ring resonators with coupling strength κ, while ring A undergoes dynamic modulation in the form of J(t) described by Eq. (1). b For an individual ring, ring A (B) supports equally-spaced resonant modes labeled as An (Bn) at frequency ωn. c After the effective coupling between two rings, the resonant modes in the system split into two symmetric and antisymmetric supermodes located at ωn ± κ. d The discrete supermodes in ring A are coupled by bichromatic modulation J(t), which can be mapped into a 1D photonic SSH model along the frequency dimension. The supermodes Cn and Dn act as synthetic lattice sites, with alternating hopping amplitudes g1 and g2. e–h The analytical band structures of the synthetic SSH model in (d) under different hopping strengths g1 and g2. Inserted: the corresponding traces of Re(G) and Im(G) as the wave vector evolving through the first Brillouin zone

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From Eq. (3), we show the capability of constructing a lattice with nonuniform connectivities between antisymmetric and symmetric supermodes Dn and Cn with alternating intra-cell and inter-cell hopping strengths g1 and g2, which is mathematically equivalent with the conventional 1D spatial SSH model but along the frequency axis of light (f) as shown in Fig. 1(d). Here, cn and dn (\(c_n^{\dagger}\) and \(d_n^{\dagger}\)) denote the annihilation (creation) operators for the supermodes Cn and Dn, which hold resonant frequency ωn + κ and ωn − κ, respectively. With definition \(\tilde c_n = c_ne^{i\left( {\omega _n + \kappa } \right)t}\)and \(\tilde d_n = d_ne^{i\left( {\omega _n - \kappa } \right)t}\), Eq. (3) can be transformed to a time-independent Hamiltonian

$$H_{{{{\mathrm{RWA}}}},I} = \mathop {\sum}\limits_n {\left( {g_1e^{ - i\phi _1}\tilde c_n^{\dagger} \tilde d_n + g_2e^{ - i\phi _2}\tilde d_n^{\dagger} \tilde c_{n - 1} + h.c.} \right)}$$
(4)

We note that the synthetic SSH lattice here in the frequency dimension is infinite if we ignore the gradual change of the FSR and the resulting off-resonance coupling between modes induced by the group velocity dispersion of the fiber35,59. Given the fact that there is no hard boundary in the frequency dimension, the infinite synthetic lattice makes the choice of intra-cell and inter-cell hoppings can be arbitrary, and different choices can lead to distinct results. To definitively simulate the topology in a SSH lattice, we therefore strict to the lattice where a pair of supermodes Dn and Cn builds a unit cell, so the connection between Cn and Dn+1 is inter-cell. The topology is then well-defined in this physical picture. We emphasize that one can certainly choose the configuration of the SSH lattice other way around, which gives an opposite topology. However, once the SSH configuration is fixed and the unit cell is defined, all the following analysis is consistent and one is possible of extracting the topological invariant in experiments.

We transfer Eq. (4) into the kf space and obtain the Hamiltonian of the synthetic lattice as \(H_{k_f} = \left( {\begin{array}{*{20}{c}} 0 & G \\ {G^ \ast } & 0 \end{array}} \right)\), where \(G = \left| G \right|e^{i\varphi \left( {k_f} \right)} = g_1e^{ - i\phi _1} + g_2e^{ik_f{\Omega} + i\phi _2}\) and φ(kf) = arg(G) being the argument of G. Here, kf denotes the wave vector reciprocal to the frequency dimension33,60. The topology of the synthetic SSH model can be characterized by the Zak phase following the definition of its spatial counterpart19

$$\varphi _{{{{\mathrm{Zak}}}}} = \frac{1}{2}{\int}_{ - \pi }^\pi {\frac{{\partial \varphi \left( {k_f} \right)}}{{\partial k_f}}} dk_f$$
(5)

The Zak phase takes two values, which are φZak = π for the topologically non-trivial case (g1 < g2) and φZak = 0 for the trivial case (g1 > g2). The system degrades to a 1D uniform lattice under the condition g1 = g2, which is thus not considered in the following discussion. The corresponding band structures read

$$\varepsilon _{k_f,m} = m\sqrt {g_1^2 + g_2^2 + 2g_1g_2\cos \left( {k_f{\Omega} + \phi _1 + \phi _2} \right)}$$
(6)

with eigenstates \(| {\psi _{k_f,m}} \rangle = ( {\psi _{k_f,m}^C,\,\psi _{k_f,m}^D} )^T = ( {1,\,me^{ - i\varphi }} )^T/\sqrt 2\) and m = ±1. \(\psi _{k_f,m}^C\) and \(\psi _{k_f,m}^D\) are the projections of eigenstates on two supermodes \(C_{k_f}\) and \(D_{k_f}\) in the kf space. Note that φ is a simple notation for the function φ(kf).

Equation (6) indicates that the band structures compose of one upper (m = 1) and one lower bands (m = −1) as shown in Fig. 1(e)–(h). The trajectories of the real part and imaginary part of G as the wave vector sweeping though the first Brillouin zone (kfΩ = 0 → 2π) are inserted in Fig. 1(e)–(h), which form circles in the Re(G) − Im(G) plane, centered at (g1, 0) with radius g2. The topology of the system can also be captured by seeing whether the circle passes through the origin, characterized by the winding number (\({{{\mathcal{W}}}} = \varphi _{{{{\mathrm{Zak}}}}}/\pi\)), where \({{{\mathcal{W}}}} = 0\) denotes the trivial case (g1 > g2) and \({{{\mathcal{W}}}} = 1\) denotes the non-trivial case (g1 < g2). Equation (6) shows that the band structure is invariant under the exchange of intra-cell and inter-cell coupling strengths g1 and g2, although this exchange operation changes the winding number and the Zak phase, moving the lattice from the topologically non-trivial regime to the trivial regime [take Fig. 1(e) and (f) for example]. This invariance or symmetry has made it challenging to distinguish the two topological phases in bulk SSH lattices without looking at edge effects30. We will show later how we overcome this challenge by measuring the optical transmission spectra and obtain the time-resolved projected band structure40,60, which is the key idea to determine the Zak phase from the bulk band structure in this simulated synthetic SSH lattice.

Projected band structure from transmission spectra

To implement our proposal, we use fibers to form two rings in the experiment, which are coupled through a 2 × 2 fiber coupler as shown in Fig. 2 [see Materials and methods]. After calibration, the lengths of both rings are 10.2 m, corresponding to a FSR of Ω = 2πvg/L = 2π·20 MHz, where vg is the group velocity and L is the length of the ring. Without modulation, we observe that the splitting distance of the two supermodes is about 2π·6.67 MHz, which gives 2κ = Ω/3 and Ω − 2κ = 2 Ω/3 [see Fig. S1 in the Supplementary Information]. To construct the synthetic SSH model, we drive the EOM by a radio frequency (RF) signal in the form of V1 cos(Ω1t) + V2 cos(Ω2t), where Ω1 = 2π·6.67 MHz, Ω2 = 2π·13.33 MHz, and V1, V2 denote the staggered modulation strengths.

Fig. 2: Experimental setup.
figure 2

Ring A and ring B are coupled by a 2 × 2 fiber coupler. EOM: electro-optic phase modulator. SOA: semiconductor optical amplifier. PC: polarization controller. DWDM: dense wavelength division multiplexing. AWG: arbitrary waveform generator. EDFA: erbium-doped optical fiber amplifier. PD: photodiode

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One can obtain the projected band structure of the synthetic lattice following several steps through the time-resolved band structure spectroscopy40,60, as we briefly summarize here. First, one collects the drop-port transmission spectrum through the output fiber coupler via linearly scanning the frequency of the input laser source. If the input laser source at frequency ω is only detuned around the reference frequency ω0 + κ with detuning ∆ω+κ ≡ ω − (ω0 + κ), one can obtain the output field around the symmetric supermode (\(S_{{{{\mathrm{out}}}}}^{ + \kappa }\)) as [see Eqs. (S1)–(S9) in the Supplementary Information]

$$S_{{{{\mathrm{out}}}}}^{ + \kappa } = - i\frac{{\gamma _{{{\mathrm{A}}}}}}{2}S_{{{{\mathrm{in}}}}}e^{ - i\omega t}\mathop {\sum}\limits_{m = \pm 1} {\left. {\frac{{\psi _{k_f,m}^{C \ast }\left( {\psi _{k_f,m}^C + \psi _{k_f,m}^De^{2i\kappa t}} \right)}}{{{\Delta}\omega ^{ + \kappa } - \varepsilon _{k_f,m} + i\gamma }}} \right|_{k_f = t}}$$
(7)

where Sin is the input laser source, γ is the total loss, and γA is the coupling strength between waveguides and ring A. Note that the reference frequency ω0 (i.e., the 0th resonant mode in a ring) is chosen dependent on the scanning frequency of the input laser source. Similarly, if the input source is detuned around the reference frequency ω0 − κ with detuning ∆ω−κ ≡ ω − (ω0 − κ), the corresponding output filed around the antisymmetric supermode (\(S_{{{{\mathrm{out}}}}}^{ - \kappa }\)) is

$$S_{{{{\mathrm{out}}}}}^{ - \kappa } = - i\frac{{\gamma _{{{\mathrm{A}}}}}}{2}S_{{{{\mathrm{in}}}}}e^{ - i\omega t}\mathop {\sum}\limits_{m = \pm 1} {\left. {\frac{{\psi _{k_f,m}^{D \ast }\left( {\psi _{k_f,m}^Ce^{ - 2i\kappa t} + \psi _{k_f,m}^D} \right)}}{{{\Delta}\omega ^{ - \kappa } - \varepsilon _{k_f,m} + i\gamma }}} \right|_{k_f = t}}$$
(8)

The superscripts ±κ label the two excited cases around ω0 ± κ separately. Here, the wave vector kf serves as a time variable33,60, due to the discrete translation symmetry along the synthetic frequency dimension. At time t, the output field is determined by the eigenvalues and eigenstates at kf = t [see Supplementary Information]. For a fixed detuning Δω±κ, the normalized output transmission (\({|{S}_{{{{\mathrm{out}}}}}^{ \pm \kappa }/{S}_{\mathrm{in}}|^2}\)) has two peaks at \({\Delta}\omega ^{ \pm \kappa } = \varepsilon _{k_f,m}\). Therefore, one can break the transmission spectrum into time slices with time window 2π/Ω1, and then stack up these time slices as a function of the input frequency detuning ∆ω, which reveals the time-resolved band structures of the system. Equations (7) and (8) indicate that if one excites ring A around ω0 + κ and ω0 − κ subsequently by sweeping the input frequency within one free spectral range, two groups of asymmetric projected band structures separated by 2κ along the frequency dimension can be acquired, which results from the projections of the band structures onto the superposition of modes \(C_{k_f}\) and \(D_{k_f}\). Note that each group of projected band structures contains two bands, and the envelope of each band is determined by Eq. (6).

In the experiment, we consider the condition of V1 < V2, and plot the projected band structures in Fig. 3(a), (b). When V1 = 1 V and V2 = 3 V are applied [see Fig. 3(a)], two groups of bands are observed at frequency detunings Δω = Ω/6, while each group contains two dispersive bands with splitting distance determined by modulation strengths. Different from the previous work40, here the time window to break the transmission spectrum 2π/Ω1 equals to three roundtrip time of ring A (3·2π/Ω) due to the superposition terms in Eqs. (7) and (8), which gives the periodicity of the projected band structure kfΩ/2π ∈ [0, 3]. We also show the projected band structure for the condition with V1 = 0 V and V2 = 3 V in Fig. 3(b), corresponding to the special case of g2/g1 → ∞ in the SSH model, which causes the four dispersive bands in Fig. 3(a) turning into four flat bands. In addition, we exhibit the simulated bands from Eqs. (7), (8) in Fig. 3(c), (d) under hopping amplitudes g1 = 0.02 Ω, g2 = 0.06 Ω and g1 = 0, g2 = 0.06 Ω, respectively, which agree with experimental results.

Fig. 3: Projected band structures under condition of V1 < V2.
figure 3

Experimentally observed projected band structures with modulation amplitudes a V1 = 1 V, V2 = 3 V, and b V1 = 0 V, V2 = 3 V. Red and blue arrows indicate the chosen bands to apply the Zak phase extraction method. Projected band structures from simulations under coupling strengths c g1 = 0.02 Ω, g2 = 0.06 Ω, and d g1 = 0, g2 = 0.06 Ω, with ϕ1 = ϕ2 = 0

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Zak phase extraction method

Here, we introduce a data-analysis scheme called resonant method to extract the important topological phases from the projected band structures in Fig. 3. By further simplifying Eqs. (7), (8) with eigenstates \(| {\psi _{k_f,m}}\rangle\), we obtain

$$S_{{{{\mathrm{out}}}}}^{ + \kappa } = - i\frac{{\gamma _{{{\mathrm{A}}}}}}{4}S_{{{{\mathrm{in}}}}}e^{ - i\omega t}\mathop {\sum}\limits_{m = \pm 1} {\left. {\frac{{1 + me^{2i\kappa t - i\varphi }}}{{{\Delta}\omega ^{ + \kappa } - \varepsilon _{k_f,m} + i\gamma }}} \right|_{k_f = t}}$$
(9)
$$S_{{{{\mathrm{out}}}}}^{ - \kappa } = - i\frac{{\gamma _{{{\mathrm{A}}}}}}{4}S_{{{{\mathrm{in}}}}}e^{ - i\omega t}\mathop {\sum}\limits_{m = \pm 1} {\left. {\frac{{1 + me^{ - 2i\kappa t + i\varphi }}}{{{\Delta}\omega ^{ - \kappa } - \varepsilon _{k_f,m} + i\gamma }}} \right|_{k_f = t}}$$
(10)

from which one sees that the phase φ [φ(kf)] of the eigenstates is actually printed in the output field, i.e., the topological phase information being encoded in the time-resolved band structure spectroscopy.

We choose one band from Fig. 3(a) indicated by the red arrow (corresponding to m = 1) as an example and re-plot it in Fig. 4(a). By taking the input frequency resonant with the eigenvalue of the chosen band at each kf, one gets \({\Delta}\omega ^{ \pm \kappa } = \varepsilon _{k_f,m}\) to obtain the most significant contribution to the output signal and the phase information from only the chosen band. An intensity parameter \(S^2 = 8\gamma ^2\left| {S_{{{{\mathrm{out}}}}}^{ \pm \kappa }} \right|^2/\left( {\gamma _{{{\mathrm{A}}}}^2\left| {S_{{{{\mathrm{in}}}}}} \right|^2} \right) = \left[ {1 + m\cos \left( {2\kappa t - \varphi } \right)} \right]_{k_f = t}\) is defined from Eqs. (9), (10) by minimizing the denominator for the chosen m band and neglecting the contribution from the other band, which can be acquired by taking the maximum output intensity value of each vertical slice (fixed kf) from the chosen band. The obtained value S2 is further normalized to the range of [−1, 1] by using the transform \(2S^2/S_{\max }^2 - 1\) as shown in Fig. 4(a), which is then used for decoding the argument φ(kf). Following this line, the argument phase φ(kf) is deduced as

$$\varphi \left( {k_f} \right) = 2\kappa t \pm \arccos \left. {\left[ {m\left( {\frac{{2S^2}}{{S_{\max }^2}} - 1} \right)} \right]} \right|_{k_f = t}$$
(11)

with periodicity of φ(kf) = φ(kf + 2π/Ω) determined by Eq. (6). Both the positive and negative values of the inverse cosine function should be considered, so that one can get the full range of the argument within the range of [−π, π]. The extracted results are shown in Fig. 4(c), where the orange and green circles represent the acquired pairs of arguments for taking positive and negative inverse cosine values, respectively. Afterwards we select out one argument value from each pair that satisfies the periodicity of φ(kf) along the horizontal direction, i.e., 2π/Ω, which is shown by the blue circles in Fig. 4(e). We then calculate the Zak phase based on Eq. (5) and acquire \(\varphi _{{{{\mathrm{Zak}}}}}^{\exp } \approx 0.98\pi\) (\({{{\mathcal{W}}}} \approx 1\)). We emphasize that characterizations of the Zak phase from the other three bands in the experiment give the same results. We also demonstrate the Zak phase decoding process for the special case V1 = 0 V and V2 = 3 V by choosing a lower band labeled by the blue arrow (m = −1) in Fig. 3(b) and show the results in Fig. 4(b), (d) and (f). The flat band leads to linearly varied arguments with wave vector [see Fig. 4(f)] and a Zak phase of \(\varphi _{{{{\mathrm{Zak}}}}}^{\exp } \approx 0.975\pi\) (\({{{\mathcal{W}}}} \approx 1\)). As comparison, we perform calculations on theoretical arguments directly from G in Fig. 4(e), (f) (the red lines), which result in φZak = π in both cases. The slight discrepancy between the theoretical and experimental values of the Zak phase is mainly caused by the insufficient accuracy of experimental equipment, the inevitable loss from the system, and the disturbance of the environment, which also cause the extracted phase not exactly periodic in experiments.

Fig. 4: Extraction process of the Zak phase under condition of V1 < V2.
figure 4

a, b The normalized maximum intensities S2 of each vertical slice (the lower panels) extracted from the chosen bands (the upper panels) in Fig. 3(a) (red arrow) and Fig. 3(b) (blue arrow), respectively. c, d The corresponding calculated arguments φ(kf) from (a) and (b) based on Eq. (11). e, f The selected arguments (blue circles) that satisfy the periodicity of 2π/Ω from (c) and (d), in comparison with the theoretical values of arg(G) (red lines), respectively

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The topology of a SSH lattice changes from non-trivial to trivial if the ratio between two hopping strengths is flipped. To show this effect, we perform the Zak phase measurement under the condition of V1 > V2 corresponding to the trivial case, where projected band structures in experiments are shown in Fig. 5(a) for V1 = 3 V, V2 = 1 V and in Fig. 5(b) for V1 = 3 V, V2 = 0 V, respectively. The system also exhibits two asymmetric groups of dispersive bands separated by 2κ = Ω/3 in Fig. 5(a), which evolves to flat bands in the limit g2/g1 → 0 [see Fig. 5(b)]. The corresponding projected band structures from simulations are plotted in Fig. 5(c), (d). One can see a good fit between the simulation results and the experimental measurements. Then, we extract the argument phases from the chosen bands in Fig. 5(a), (b) by following the same procedure in Fig. 4 and show the results in Fig. 5(e), (f) (blue circles), which agree well with the theoretical values calculated from G (the red lines). The corresponding Zak phases are integrated as \(\varphi _{{{{\mathrm{Zak}}}}}^{\exp } \approx 0.05\pi\) and \(\varphi _{{{{\mathrm{Zak}}}}}^{\exp } \approx 0.03\pi\) (\({{{\mathcal{W}}}} \approx 0\)), respectively, while the theoretical arguments give ϕZak = 0. Therefore, we show that our system has the capability for directly decoding the topological phase by analyzing the projected band structure from the transmission spectra.

Fig. 5: Measurements of the Zak phase under condition of V1 > V2.
figure 5

Experimental observed band structures with modulation amplitudes a V1 = 3 V, V2 = 1 V, and b V1 = 3 V, V2 = 0 V. Band structures from simulations under coupling strengths c g1 = 0.06 Ω, g2 = 0.02 Ω, and d g1 = 0.06 Ω, g2 = 0, with ϕ1 = ϕ2 = 0. e, f Measured arguments (blue circles) extracted from the chosen bands in (a) (red arrow) and (b) (blue arrow), compared with theoretical arguments calculated by G (red lines), respectively

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Discussion

Besides extracting the Zak phase from the bulk band structure, one can also distinguish the topology of the system from the distinct band shapes in the topologically non-trivial case (Fig. 3) and in the trivial case (Fig. 5), which exhibit very different patterns [also see Fig. S2 in the Supplementary Information]. The band shapes in Fig. 3 show characters of shorter segments compared to those in Fig. 5. Unfortunately, the proposed Zak phase extraction method using the intensity parameter S2 is not applicable for cases with smaller band gap compared to the total loss γ including the gapless case at the phase transition point with V1 = V2. The key finding of the synthetic SSH model here originates from the frequency difference between the lattice sites in the frequency dimension, which causes the superposition of two supermodes and the topological phase information thus encoded in the bulk band. In addition, the topologically non-trivial and trivial transmission spectra can be converted flexibly by the external modulation, which brings possible ingredient towards exploring spectral non-reciprocity in the future61,62,63, and hence might bring potential opportunity in achieving the active optical isolator and circulator operated at the telecom wavelength64,65,66. Our proposed method for characterizing the topological invariant is universal in the synthetic frequency dimension, and can be used for studying topology with long-range couplings67,68. The ability of providing alternating modulations between synthetic lattice sites in experiment therefore offers new possibility to construct more complex lattice structures with nonuniform connectivities in the frequency dimension44,45,46,47,48,49. Our approach in fiber-based ring system can be extended to the microring resonators due to the advance of on-chip integrated photonics69,70. Moreover, our results exhibit great potentials in linking towards further connecting photonic molecules52, and hence hold applications in photonic computation and quantum systems71,72.

In summary, we implement the experimental measurement of topological Zak phase in a synthetic SSH model by utilizing the frequency axis of light, constructed by two coupled ring resonators modulated by bichromatic signals of different amplitudes. We find that the signatures of the topological invariant characterized by the Zak phase are imprinted in the distinguishable time-resolved transmission spectra, which can be extracted by the proposed resonant method. Quantized Zak phases are observed for non-trivial and trivial cases, where experimental measurements show excellent agreement with the results in theory. The main advantage of this method is to extract the topological invariant directly from the bulk band structures. With appropriate designs, the spin or valley degree of freedom is possible to be included in models in the synthetic frequency dimension37,38, and hence the corresponding topological invariant may also be directly captured. Our work provides the evidence of directly reading the topological phase from the bulk band structure and paves a new route to explore topological phases in higher-dimensional topological materials51,73,74,75. For example, one way to construct higher-dimensional lattices in the synthetic space is to add long-range connectivities, which can be used to build higher-dimensional synthetic space76,77. The superposition of modes along two directions may be obtained so higher-dimensional topological phases can be further studied. We anticipate that future researches in this synthetic frequency platform may find potential applications with physical phenomena in high dimensions and even bulk-defect responses13 with reconfigurability, scalability, and flexibility.

Materials and methods

The two fiber rings are coupled through a 2 × 2 fiber coupler with coupling ratio 50:50. We excite ring A by a tunable laser source (linewidth of 200 kHz) centered at 1550.92 nm, which can be finely scanned over 30 GHz by applying a ramp signal to its frequency modulation input module. A 2 × 2 fiber coupler with coupling ratio 99:1 couples 1% of the laser source to ring A, and a second 99:1 fiber coupler couples out 1% of the signal into the output waveguide for detection. A lithium niobate EOM is placed inside ring A driven by an arbitrary waveform generator (200 MHz bandwidth). The polarization controller is used to calibrate the polarization of resonant frequency modes circling in each ring to the principal axis of EOM. To achieve a high quality factor, a semiconductor optical amplifier is utilized for compensating the losses in the ring, where the amplified spontaneous emission noise is filtered by a dense wavelength division multiplexing centered at 1550.92 nm (international telecommunication union channel 33, 100 GHz bandwidth). The output signal from ring A is amplified by an erbium-doped optical fiber amplifier before being sent to a fast InGaAs photodiode (10 GHz bandwidth) for detection and is then sent to the oscilloscope (5G samples/s with 1 GHz bandwidth) for analysis. To guarantee the two rings having the identical lengths, we also put an unmodulated EOM, and two 2 × 2 fiber couplers with coupling ratio 99:1 in ring B, which are not plotted in Fig. 2.