In the past ten years, the study of topological facies has expanded from condensed matter physics to many different branches of physics, including artificial physical systems, which has stimulated extensive research on topological phases in various fields. The study of synthetic dimensions is to simulate the movement of particles in the crystal lattice by coupling the inherent degrees of freedom of the system to construct an artificial crystal lattice. There are many degrees of freedom that can be used in photonic systems, such as the frequency of photons, orbital angular momentum, space mode, and timing of pulses. The synthetic dimension enables researchers to study high-dimensional physical problems in relatively simple low-dimensional optical space, greatly simplifies the structural design of devices, and provides more sophisticated and diverse means of manipulating the light field.

In recent years, the synthetic frequency dimension constructed based on resonant ring has attracted extensive attention from researchers. By applying electro-optical phase modulation to dynamically adjust the permittivity of the resonant ring, resonant modes at different frequencies can be coupled. Flexible additive modulation can provide specific connections to build rich and diverse lattice models, enabling physical phenomena that are difficult to achieve in real space, and have experimental reconfigurability.

The synthetic dimension provides a powerful topological photonics research platform, and how to characterize, implement, and detect topological phases is still the core problem. The different topological phases of a system can be characterized and classified using topological invariants.

In one-dimensional systems, the Su-Schrieffer-Heeger (SSH) model can be characterized using the Zak phase as a topological invariant. The researchers proposed and realized several topological photonics experimental schemes for measuring the phase of Zak, including Bloch oscillation combined with Ramsay interferometer, average displacement of photonic quantum walking, leakage mode photonic lattice, absorption spectrum of atomic gas, etc.

However, in existing photonics or condensed matter physics platforms, band topological information cannot be obtained directly from the band measurements of the usual 1D SSH model. The two phases of topological mediocre and topological non-trivial have the same band structure, and it is necessary to measure the Zak phase or boundary effect by additional experimental means to identify them. The construction of SSH lattice in the synthetic frequency dimension based on resonant loop can bring richer physical phenomena and provide a new way for topological phase resolution.

Recently, Chen Xianfeng’s research group, Yuan Luqi’s research group and collaborators in the School of Physics and Astronomy of Shanghai Jiao Tong University have made new progress in the field of synthetic dimension, and constructed an artificial synthetic SSH model in the frequency dimension of two coupled resonant rings by using dual-frequency modulation, and for the first time experimentally verified the direct extraction of Zak phase in the measurement of energy band structure. This research result combines topological photonics research and synthetic band analysis in frequency dimension, and proposes a new method to directly extract topological invariants by using time-resolved band measurement.

The results were published in Light: Science & Applications under the title “Direct Extraction of Topological Zak Phase with the Synthetic Dimension.”

Assistant researchers Li Guangzhen and Wang Luojia of Shanghai Jiao Tong University are co-first authors of this paper, special researcher Yuan Luqi and Professor Chen Xianfeng are co-corresponding authors, and collaborators include Shanghai Jiao Tong University graduate students Ye Rui and Zheng Yuanlin, Zhejiang University Professor Wang Dawei, Peking University Professor Liu Xiongjun, and University of Maryland Professor Avik Dutt. This research work has been supported by the National Natural Science Foundation of China, the National Key Research Program, and the Shanghai Natural Science Foundation of China.

The research team uses the coupling of two resonant rings to generate antisymmetric and symmetric modes, the resonant frequency is ω0+nΩ±κ (ω0 and Ω are the central frequency and free spectral range of the A ring or B ring resonance mode, κ is the coupling intensity between the two rings), forming unequally spaced lattice points in the synthetic frequency dimension. Applying dual-frequency electro-optical modulation to the A-ring at frequencies of 2κ and Ω-2κ can produce coupling coefficients g1 and g2 alternately arranged between the grids of the SSH model.

Therefore, the synthetic SSH model is constructed in the synthetic frequency dimension, as shown in Figure 1(a). Under different topological phases, the SSH model has the same band structure, while the Zak phase is different. The phase obtained by the eigenstates during the adiabatic evolution of the momentum space is shown in Figure 1(b), and the Zak phase is defined as 1/2 of this value.

**Figure 1 (a) SSH model constructed in the synthetic frequency dimension of two coupled resonators under electro-optical modulation; (b) Adiabatic phase evolution of computational Zak phases in topological non-trivial (left) and topological trivial (right) phase energy band structures and momentum spaces.**

Experimentally, the research team used two fiber optic rings with a length of 10.2 meters to build a resonant loop system, as shown in Figure 2. Without modulation, the frequency spacing between the modes split caused by the coupling of the two loops is about 2κ=Ω/3=2π·6.67MHz. The strength and phase of the dual-band modulated signal can be flexibly controlled. An optical signal is input and output from the A-ring. The frequency of the incident light is linearly modulated near the resonant frequency of the resonant ring. The transmittance signal at the output is segmented according to a time window of 2π/2κ, and the range of each small transmittance after splitting is approximately considered to correspond to an input frequency detuned Δω.

**Figure 2 Experimental setup.**

Each small segment of the split transmittance is stacked to obtain a time-resolved projection band map of the system. As shown in Figure 3, there are two asymmetrical bands at frequencies spaced 2κ apart. The two energy bands in each group correspond to the two energy bands of the SSH model, and the transition of the topological phase is determined by the modulation intensity.

**Figure 3: Evolution of phase φ obtained by experimentally measured band structure and Zak phase extraction in momentum space; (a) Topological non-trivial phase g1****g2；**； (b) Topological mediocrity phase g1 >**(b) Topological mediocrity phase g1 > g2.**

Theoretical calculations show that the change in transmittance signal intensity on the band comes from the projection of the band structure on the superposition states of symmetric and antisymmetric Bloch modes. Since the transmittance signal requires a superposition of symmetry and antisymmetry modes, the phase φ of the Bloch-Hamiltonian eigenstates of the SSH model is encoded into the band structure. The evolution of this phase in momentum space is used to calculate the Zak phase, as shown in Figure 1(b). Therefore, the phase φ can be reversed from the strength distribution of the time-resolved band signal, so as to directly extract the Zak phase of the SSH model. The measured band structure shown in Figure 3 corresponds to different topological phases. The evolution of the obtained phase φ is consistent with the theoretical results of the energy band shown by the arrow, and the topological phase of the system can be distinguished. As shown in Figure 3(a), when modulation intensity g1 < g2, the phase φ evolves from 0 to 2π, and the Zak phase is calculated to be π. As shown in Figure 3(b), when modulating intensity g1 >g2, the evolution of the phase φ does not bypass the origin, and the Zak phase is calculated to be 0.

This work is the first experimental measurement of topological phase-resolved SSH band structures in the synthetic frequency space based on resonant rings, and the extraction of Zak phases is realized directly from the signal transmission spectrum. In the synthetic frequency dimension, this method of extracting topological invariants is universal, and provides a simple and feasible experimental approach with reconfigurability for studying topological phases. Experimentally, the alternating coupling strength between synthetic lattices is realized, which lays a solid foundation for constructing more complex lattices by using the synthetic frequency dimension in the future. The ability to flexibly convert topological phases in simultaneous modulation resonant loop systems has potential applications in optical asymmetry and optical communications. (Source: LightScience Applications WeChat public account)

Related paper information:https://doi.org/10.1038/s41377-023-01126-1

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