Realizing topological edge states with Rydberg-atom synthetic dimensions

The study explores synthetic dimensions—discrete internal or external states engineered to emulate motion in a real-space lattice—to realize and probe topological band structures. Building on prior demonstrations using motional, spin, rotational, and photonic degrees of freedom, the authors harness Rydberg levels of 84Sr atoms coupled by millimeter waves to create a controllable synthetic lattice. The central research goal is to realize the Su-Schrieffer-Heeger (SSH) model in a synthetic spatial dimension and directly observe symmetry-protected topological edge states, characterize their localization and energies, and test their robustness to disorder that preserves or breaks chiral symmetry. This platform promises high connectivity control, large Hilbert spaces, and tunable interactions, enabling configurations inaccessible in real space and offering new avenues for quantum simulation.

The paper situates its work within extensive prior research on synthetic dimensions and topological matter. Atomic platforms have realized synthetic dimensions using Raman-coupled ground-state sublevels and optical clock transitions, observing artificial gauge fields, spin-orbit coupling, and chiral edge states. Momentum-state lattices coupled by Bragg transitions enabled studies of Anderson localization, artificial gauge fields, and SSH topology. Proposals for molecular rotational-state synthetic dimensions are closely related. Reviews on topological band theory and topological insulators establish the broader context. Rydberg systems add strong, tunable dipole-dipole interactions and many accessible levels, enabling larger synthetic spaces and potentially local interactions in synthetic space, which are difficult in other platforms. The synthetic-dimension perspective has also been applied to Rydberg molecules, e.g., conical intersections.

• Atom preparation: Ultracold 84Sr atoms are produced via two-stage magneto-optical trapping and loaded into a 1064 nm crossed-sheet optical dipole trap. After forced evaporation, samples contain ~1e5 atoms at peak density ~1e11 cm−3 and temperature ~2 μK. A 4 G magnetic field defines quantization and Zeeman-splits sublevels by ~11 MHz.
• Synthetic lattice construction: Six Rydberg levels (three ns 3S1(m=1): 57s, 58s, 59s mapped to sites i=1,3,5; and three np 3P0 levels mapped to i=2,4,6) are near-resonantly coupled by multiple millimeter-wave frequency components to form a one-dimensional chain with alternating strong and weak tunneling. The effective lattice Hamiltonian in the rotating-wave frame is a tight-binding chain with nearest-neighbor tunneling amplitudes Ji+1 = Ωi+1/2 set by microwave Rabi amplitudes and on-site potentials δi set by microwave detunings.
• SSH configurations: Topological configuration (with edge states): weak tunneling at the boundaries (i=1–2 and 5–6) and strong tunneling on interior bonds; trivial configuration exchanges weak/strong at edges. Strong coupling Js varied 0.5–1.5 MHz; weak coupling Jw = 100 kHz, giving Js/Jw = 5–15. Millimeter-wave frequencies are adjusted to keep δi≈0 unless intentional disorder is introduced.
• Probing band structure: Two-photon excitation from the ground state via the 5s5p 3P1 intermediate (detuned by +80 MHz) selectively excites ns sites (odd i). A 5 μs pulse is used. Rydberg excitation spectra versus probe detuning provide peaks at the eigenenergies whose weights reflect overlaps |⟨β|ip⟩|^2 with the chosen site. Spectral features are fit with sinc-squared lineshapes (from 5 μs pulse) convolved with a 100 kHz Gaussian (laser and natural broadening) to extract peak positions and areas.
• Site-resolved detection: Selective field ionization (SFI) uses an electric-field ramp E(t)=E0(1−e^{−t/τ}) with E0≈49 V/cm and τ=6.5 μs to map Rydberg levels to ionization times, yielding site populations (ns vs np/(n+1)s are partly unresolved in current setup but allow two-site resolution).
• Data analysis: Band structures and eigenstate decompositions are compared to exact diagonalization of the 6-site SSH Hamiltonian. Additional studies vary tunneling imbalance (±15%) and on-site potentials (by detuning specific couplings) to probe robustness and chiral symmetry breaking.
• Theory and decoherence modeling: Lindblad master-equation simulations for the driven lattice-plus-probe system are performed with models including no decoherence and amplitude-noise-induced decoherence on millimeter waves. Decoherence rates Γ_{ipr} are taken proportional to the associated bond strength, Γ_{ipr}=C J_{ipr+1}, and are fit to spectra. Off-resonant sublevels and counter-rotating terms are analyzed and found negligible for lineshape broadening under experimental conditions.

• Realization of SSH band structure: Spectra exhibit two bulk bands separated by a gap, with bulk-state features at detunings ≈ ±Js and a mid-gap feature at zero detuning corresponding to edge states. Peak positions versus Js agree with exact diagonalization of the 6-site SSH Hamiltonian.
• Observation of topological edge states (TPS): For the topological configuration (weak boundary bonds), a strong mid-gap signal appears at Δ≈0, indicating edge modes centered in the bandgap. The gap width is ~2Js. The edge-state spectral weight is largest when probing the boundary ns site (57s, i=1), smaller for 58s (i=3), and minimal for 59s (i=5), matching calculated overlaps |⟨β|i⟩|^2.
• State localization and decomposition: SFI confirms that excitation at the edge-state peak yields electrons predominantly from the 57s boundary site (i=1), whereas excitation at the bulk peaks yields population on bulk sites (i=2–5). Spectral areas under edge and bulk features quantitatively match the summed squared overlaps from exact diagonalization, with edge contributions summing to one and bulk to two across probed sites, consistent with two edge and four bulk states and probe sensitivity only to odd sites.
• Dependence on coupling ratio: Increasing Js/Jw from 5 to 15 enhances edge-state localization on the boundary site and increases bulk splitting proportionally to Js, in agreement with theory.
• Trivial configuration: Exchanging weak/strong bonds at the edges eliminates mid-gap states; spectra show only bulk features with no in-gap peaks, and calculated decompositions show all states have bulk character.
• Robustness to tunneling imbalance preserving chiral symmetry: Introducing ±15% imbalance separately on strong bonds J2–3 or J4–5 significantly modifies bulk-state splittings localized near the affected sites, while leaving the edge-state energy pinned at zero, demonstrating symmetry protection.
• Chiral-symmetry breaking via on-site potentials: Detuning the 57s–57p coupling (shifting the i=1 on-site energy) shifts the edge-state energy away from zero; detuning the 59s–59p coupling (shifting i=6) leaves the probed edge-state energy unchanged, confirming the probed edge mode is localized on i=1 and illustrating that on-site terms break chiral symmetry and unpin edge energies.
• Decoherence characterization: Master-equation fits without decoherence reproduce most spectral features; incorporating amplitude-noise-induced decoherence captures residual linewidth broadening and tail smoothing. Extracted decoherence scales with coupling strength; notably, Γ for ipr=5 is ~3× larger than other sites in the topological configuration, suggesting technical noise or spurious couplings. Coherence times at low coupling are consistent with narrow (~50 kHz) linewidths in isolated two-level spectroscopy, indicating decoherence grows with microwave drive strength.

The experiments demonstrate that a synthetic spatial dimension formed by coupled Rydberg levels can faithfully implement a canonical topological lattice model (SSH) and reveal its symmetry-protected edge modes. By measuring photoexcitation spectra and SFI-resolved populations, the authors directly access both the band structure and eigenstate composition, showing edge localization and zero-energy pinning as predicted by chiral symmetry. Perturbation studies distinguish symmetry-preserving tunneling imbalance, which leaves edge energies invariant, from chiral-symmetry-breaking on-site potentials, which shift edge modes. These results validate Rydberg-atom synthetic dimensions as a versatile and highly controllable quantum-simulation platform. The approach benefits from large state manifolds and strong dipole couplings, enabling complex synthetic geometries; it is compatible with introducing controlled phases (gauge fields) and is well suited to dynamical studies. Importantly, Rydberg dipole-dipole interactions are naturally local in synthetic space, opening avenues to explore interacting topological phases and novel many-body phenomena that are challenging in other synthetic-dimension implementations.

The work realizes a six-site SSH model in a Rydberg-atom synthetic dimension, observes topological edge states at zero energy, and verifies their localization and symmetry protection. Band structures and state decompositions extracted from spectroscopy and SFI agree with exact diagonalization and master-equation simulations. By tuning millimeter-wave detunings, the authors introduce on-site potentials that break chiral symmetry and controllably shift edge-state energies. The results establish Rydberg synthetic dimensions as a promising platform for quantum simulation with flexible control over connectivity, tunneling, and phases. Future directions include scaling to larger and higher-dimensional synthetic lattices, engineering artificial gauge fields via controlled tunneling phases, exploring Floquet and out-of-equilibrium phenomena, and extending to interacting many-body systems using arrays of single Rydberg atoms in optical tweezers. Addressing technical decoherence, managing AC Stark shifts, and mitigating spurious couplings will further expand the accessible physics.

• Decoherence increases with stronger millimeter-wave coupling, leading to additional linewidth broadening and reduced contrast; the dominant source (e.g., amplitude fluctuations, spurious multi-photon couplings to higher-l states) is not yet conclusively identified.
• AC Stark shifts and coupling to off-resonant magnetic sublevels introduce level shifts that require calibration and compensation.
• The optical probe couples only to ns (odd) sites, limiting direct sensitivity to half the lattice sites and necessitating inference for even-site components.
• SFI resolution does not fully separate np and (n+1)s arrival times under current conditions, yielding only partial site resolution.
• Scaling to larger synthetic spaces may require multiple local oscillators and antennas to cover broader frequency ranges, adding technical complexity.
• Decoherence modeling indicates site- and bond-dependent noise not fully captured by simple proportional models, suggesting more refined noise characterization is needed for quantitative predictiveness.