Realizing topological edge states with Rydberg-atom synthetic dimensions

Synthetic dimensions, a powerful tool in quantum simulation, utilize internal or external states to mimic particle motion in a real-space lattice. This approach offers exceptional control and allows for the creation of configurations unattainable in real space, opening possibilities for higher-dimensional systems, complex topologies, and artificial gauge fields. Previous experiments have employed various degrees of freedom such as motional, spin, and rotational levels in atoms and molecules, and frequency or spatial modes in photonics to create synthetic dimensions. Atomic synthetic dimensions have demonstrated artificial gauge fields, spin-orbit coupling, and chiral edge states using Raman-coupled magnetic sublevels or single-photon-coupled electronic orbitals on a 1D optical lattice. Alternatively, discrete motional states coupled with Bragg transitions have also been used to observe phenomena like Anderson localization and topological states. This research utilizes Rydberg levels in ⁸⁴Sr atoms to realize a synthetic lattice, leveraging the mathematical equivalence between coupled Rydberg levels and particles tunneling between lattice sites. This approach allows for precise control over connectivity, tunneling rates, and on-site potentials, creating a wide array of synthetic systems, including those impossible to realize in physical space. The abundance of Rydberg levels and strong transition dipole moments enable the creation of large and intricate synthetic landscapes. Further, Rydberg dipole-dipole interactions offer a mechanism for tunable localized interactions for many-body systems in synthetic space, a significant advantage over other atom-based platforms. The goal of this study is to demonstrate the capabilities of Rydberg-atom synthetic dimensions by realizing the Su-Schrieffer-Heeger (SSH) model in synthetic space and investigating its topologically protected edge states (TPS) and their robustness to disorder. The SSH model, known for its alternating weak and strong tunneling, provides a paradigmatic example for exploring topological matter and the robustness of its edge states to perturbations.

The field of synthetic dimensions has seen significant progress, with various implementations and applications explored in recent years. The concept has been theoretically proposed and experimentally realized using various physical systems, including ultracold atoms, molecules and photons. For ultracold atoms, synthetic dimensions have been created using internal states such as spin or hyperfine levels, manipulating the coupling between these levels to simulate lattice Hamiltonians. This has allowed for the simulation of various topological phases and the observation of phenomena like chiral edge states and artificial gauge fields. In the case of molecules, rotational levels have been proposed and partially implemented to generate synthetic dimensions. Photonic systems provide another avenue for creating synthetic dimensions, with the ability to use frequency or spatial modes as synthetic dimensions. The existing literature provides ample theoretical and experimental evidence for the efficacy and versatility of synthetic dimensions in quantum simulation. This work builds upon the foundation laid by these prior studies by introducing a novel approach using Rydberg levels of strontium atoms, offering unique advantages in terms of control and interaction engineering.

The experiment uses ultracold ⁸⁴Sr atoms trapped in an optical dipole trap. Millimeter waves couple Rydberg levels |i⟩ and |j⟩ with amplitude Ω_ij, resulting in a Hamiltonian equivalent to a particle tunneling between lattice sites (i) and (j) with tunneling amplitude J_ij = Ω_ij/2. A six-site synthetic lattice is constructed using three 5sns ³S₁(m = 1) and three 5snp ³P₀ levels. The Hamiltonian, neglecting counter-rotating terms and transforming into a rotating frame, is given by: H_lattice = Σ_i=1⁵ (-ħJ_i+1|i⟩⟨i+1| + h.c.) + Σ_i=1⁶ ħδ_i|i⟩⟨i|, where J_i+1 are the tunneling amplitudes, δ_i are on-site potentials, and ħ is Planck's constant. The SSH model is realized by setting δ = 0 and alternating weak (J_w) and strong (J_s) couplings. A 4 Gauss magnetic field splits Zeeman sublevels to minimize unwanted couplings. The synthetic lattice is populated and probed using two-photon excitation from the ground state to Rydberg levels via an intermediate state, with the excitation rate described by Γ(Δ_l) = πΩ² Σ_β |⟨β|i_p⟩|²δ(Δ_l − ε_β/ħ), where Ω_i is the effective two-photon Rabi frequency, |β⟩ and ε_β are the eigenstates and eigenenergies of H_lattice, and |i_p⟩ is an unperturbed Rydberg level. Rydberg populations are detected using selective field ionization (SFI), identifying the occupied synthetic-lattice site based on electron arrival time. The band structure is probed by varying the two-photon laser detuning Δ_l, providing information about eigenstate overlaps. To analyze the robustness of the edge states, perturbations are introduced by imbalancing the strong couplings (preserving chiral symmetry) and by shifting on-site potentials (breaking chiral symmetry). Numerical simulations, using the Lindblad master equation, are performed to model the system and account for decoherence effects, particularly amplitude noise on the millimeter waves. The decoherence rates are fitted to match the observed linewidths.

The experiment successfully realized the SSH model in a synthetic dimension using Rydberg levels of ⁸⁴Sr atoms coupled by millimeter waves. The observed band structure and eigenstate decomposition closely matched theoretical predictions. Topologically protected edge states (TPS) at zero energy were observed, a hallmark feature of the SSH model. The edge states exhibited robustness against perturbations that preserved chiral symmetry, as the energy remained pinned at zero. However, the introduction of on-site potentials (by altering millimeter-wave detunings), which broke the chiral symmetry, shifted the edge state energies. Selective field ionization (SFI) measurements confirmed the localization of the edge states primarily on boundary sites. Analysis of the spectra revealed a strong correlation between the intensity of the spectral features and the overlap between the eigenstates and unperturbed Rydberg levels. The observed spectral linewidths provided insights into the decoherence mechanisms present in the system. Numerical simulations using the Lindblad master equation, incorporating amplitude noise as a model for decoherence, successfully replicated the experimental spectra with remarkable accuracy. The analysis showed a significant increase in the decoherence near the edge state for one particular probe transition. This may be due to the sensitivity of the edge state eigenfunctions or to noisy coupling in the system. These results support the use of Rydberg levels as a robust and versatile platform for constructing synthetic dimensions.

This research demonstrates the feasibility and potential of Rydberg-atom synthetic dimensions for quantum simulation. The successful observation of topologically protected edge states in the SSH model validates the ability to engineer and control complex quantum systems using this approach. The robustness of the edge states against symmetry-preserving perturbations highlights the potential for creating stable and fault-tolerant quantum devices. The detailed agreement between experimental results and theoretical simulations, including decoherence models, adds confidence to the accuracy and completeness of the model used. The quantitative insights gained from the spectral analysis and SFI measurements enhance the understanding of the underlying physics. The identification of noise sources and their relation to experimental parameters, as obtained by the fitting of the Lindblad model, will be crucial for future improvements in the system's coherence properties and the realization of more complex quantum simulations. The findings open up exciting possibilities for exploring a wide range of phenomena in higher-dimensional synthetic lattices, including interacting many-body systems.

This study successfully demonstrated Rydberg-atom synthetic dimensions as a powerful platform for quantum simulation by experimentally realizing and characterizing the topological edge states of the SSH model. The experimental results demonstrate high accuracy and precision, achieving excellent agreement with theoretical predictions, including decoherence effects. This establishes Rydberg-atom synthetic dimensions as a promising approach for studying topological quantum matter and offers a flexible platform for exploring various complex many-body systems by extending the size of the synthetic lattice and introducing interactions between sites. Future work could focus on extending the size of the synthetic lattice and investigating more complex topological phases and interacting many-body systems.

The current study is limited to a six-site SSH model. Scaling up to larger synthetic lattices might encounter challenges related to decoherence and experimental complexity. The decoherence model used in the simulations is a simplified representation of the actual noise sources. A more detailed understanding and characterization of decoherence mechanisms are necessary for achieving even higher experimental precision and control. The use of triplet Rydberg states could introduce spurious couplings to other states, as suggested by the dependence of the fitted decoherence parameter. Exploring singlet Rydberg states could potentially improve the coherence properties.