Direct extraction of topological Zak phase with the synthetic dimension

Topological photonic materials have garnered significant attention due to their unique properties, including topologically protected edge states and unidirectional light transport. These properties stem from topological invariants, which characterize the topological phases of matter. The Zak phase, a topological invariant for one-dimensional (1D) systems, is typically obtained from the number of edge states or interference patterns. However, directly extracting the Zak phase from the bulk band structure of a 1D system, specifically the SSH model, has proven challenging due to the identical shapes of bulk band structures in both trivial and non-trivial phases. This paper addresses this challenge by leveraging the synthetic frequency dimension, a powerful platform for creating lattices with artificial connectivities not readily achievable in real space. Previous work in synthetic frequency dimensions has explored various lattice structures and functionalities, including Hall ladders and non-Hermitian topologies. However, experimental implementations of lattices with nonuniform connectivities, crucial for realizing richer physics, have been limited. This research demonstrates the experimental extraction of the Zak phase directly from the bulk band structure of a 1D synthetic SSH model constructed along the frequency dimension. This is achieved by using two coupled ring resonators where the symmetric and antisymmetric supermodes are connected by an electro-optic phase modulator (EOM) that provides bichromatic sinusoidal modulations. The authors exploit the unique feature that the identical shapes of band structures are broken due to distinct projections of the band structures onto superpositions of the two supermodes. The topological phase information is encoded in the time-resolved projected band structure and is extracted from transmission spectra.

The study extensively reviews existing methods for probing the Zak phase in 1D photonic systems, including those based on Bloch oscillations, Ramsey interferometry, leaky photonic lattices, and chiral symmetry breaking. These methods, however, typically rely on edge states or interference effects and cannot directly extract the Zak phase from the bulk band structure. The authors highlight the limitations of current platforms in distinguishing between trivial and non-trivial topological phases based solely on the bulk band structure. They also discuss the use of synthetic frequency dimensions in creating artificial lattices with novel functionalities, citing previous theoretical and experimental work on Hall ladders, dynamic band structures, non-Hermitian topologies, and flat bands. The lack of experimental implementations of nonuniform connectivities in synthetic frequency dimensions is emphasized, motivating the present work.

The authors construct a 1D synthetic SSH model using two identical coupled ring resonators. The ring resonators support equally spaced resonant modes, which, upon coupling, split into symmetric and antisymmetric supermodes. An EOM, driven by a bichromatic RF signal, modulates the system, creating alternating hopping strengths between the supermodes, effectively mimicking the SSH model in the frequency dimension. The Hamiltonian of the system is derived, and the rotating wave approximation is used to simplify the analysis. The resulting Hamiltonian is then transformed into k-space, enabling the calculation of the Zak phase using its definition as an integral over the Brillouin zone. The authors demonstrate that the band structure is invariant under the exchange of intra-cell and inter-cell coupling strengths, but the winding number and Zak phase change accordingly, highlighting the challenge of distinguishing topological phases from the bulk band structure alone. To overcome this challenge, the authors employ time-resolved band structure spectroscopy to obtain the projected band structure from the transmission spectra. The experimental setup involves two coupled ring resonators connected by a fiber coupler, with an EOM placed inside one ring. The EOM is driven by a bichromatic RF signal with adjustable amplitudes. Transmission spectra are measured by scanning the input laser frequency, and the data is analyzed to extract the projected band structure. The system parameters, such as FSR, coupling strength, and modulation strengths are experimentally calibrated. A method to extract Zak phase from the projected band structure is introduced. This involves choosing a specific band from the projected band structure and identifying the input frequency that is resonant with the eigenvalue of that band at each k-point. The output signal corresponding to these resonant frequencies is then analyzed to determine the phase information related to the Zak phase. The Zak phase is obtained by analyzing the phase of the eigenstates which are printed in the output field, thus encoded in the time-resolved band structure spectroscopy. This is done by fitting the intensity of the chosen band's signal to extract the phase, φ(k). This resonant method is employed to minimize contributions from other bands and isolate the phase information of the chosen band.

The key findings include the successful experimental construction of a synthetic SSH lattice in the frequency dimension using coupled ring resonators and bichromatic modulation. The authors demonstrate that the topological phase information (Zak phase) is encoded in the time-resolved projected band structure obtained from the transmission spectra, overcoming the limitation of identical bulk band structures in trivial and non-trivial phases. Experimentally measured transmission spectra are presented, showing clear distinctions between the trivial and non-trivial topological phases. The projected band structures are extracted from these spectra, showing good agreement with theoretical simulations. The Zak phase values extracted from the experimental data (approximately 0 and 0.98π) are consistent with the theoretical predictions for trivial and non-trivial phases, respectively. The introduced resonant method for extracting the Zak phase from the projected band structure provides a direct and robust way to characterize the topological properties of the synthetic SSH lattice.

This work presents a significant advancement in characterizing topological phases of matter. The direct extraction of the Zak phase from the bulk band structure, without relying on edge states or interference effects, simplifies the experimental characterization of topological systems. The use of synthetic dimensions provides a versatile platform for manipulating and controlling topological properties. The successful implementation of the method in a fiber-based platform demonstrates its experimental feasibility and potential for integration with existing optical communication technologies. The demonstrated ability to distinguish between trivial and non-trivial topological phases based solely on bulk properties opens new possibilities for designing and implementing topological photonic devices with enhanced functionalities. The agreement between experimental results and theoretical simulations validates the accuracy and robustness of the method. This method is fundamentally different from previously reported methods, making it a unique and valuable approach to topological phase characterization.

The researchers successfully demonstrated a novel method for direct extraction of the topological Zak phase from the bulk band structure using a synthetic frequency dimension. The experimental results agree with theoretical predictions, validating the approach's accuracy. This method offers a simple and experimentally feasible route for exploring topological phases in synthetic systems and has potential applications in integrated photonics and optical communications. Future work could focus on extending this technique to higher-dimensional systems and investigating the effects of non-Hermiticity and nonlinearities.

The current experimental implementation is limited to a 1D system. While the authors suggest the method's potential extension to higher dimensions, further investigation is needed. The accuracy of the Zak phase extraction depends on the precision of the experimental measurements and the data analysis method. The assumption of ideal components and negligible noise in the experimental setup may not be entirely realistic in practical applications. The analysis relies on the rotating wave approximation, which may introduce some limitations depending on the specific system parameters.