Geometry-Induced Domain-Wall Pinning and $\mathbb{Z}_2$ Asymmetry in Nominally One-Dimensional Rydberg Arrays

AI summary

Overview

Research area: Quantum simulation with neutral-atom analog quantum computers — specifically, Rydberg-atom arrays used to emulate antiferromagnetic Ising-type spin chains, and how the physical embedding of a nominally one-dimensional chain in a two-dimensional array alters the physics.

Technical level: Intermediate. The paper assumes familiarity with the transverse-field Ising model, Rydberg blockade, and the PXP constrained model, but its central message is accessible to anyone who knows basic spin-chain physics.

Scope (one sentence): The authors run quasi-adiabatic sampling experiments on QuEra's 33-atom triangular and 47-atom square-outline Rydberg arrays to show that corner geometry and an engineered vacancy pin domain walls and break the ℤ₂ symmetry that a pure one-dimensional Ising model would require.

What This Paper Is About

Neutral-atom quantum computers such as Aquila are often used to emulate simple spin models like the antiferromagnetic Ising chain, and a natural expectation is that a chain of atoms laid out along a one-dimensional outline behaves like a one-dimensional system. But the hardware forces every layout into a two-dimensional plane, so corners, sharper-than-180° angles, and missing atoms change the longer-range van der Waals couplings in ways a pure 1D model does not contain. This paper measures exactly those deviations: it tunes the analog control parameters so the overall fraction of Rydberg (up) and ground-state (down) atoms is nearly balanced, then maps where domain walls between antiferromagnetic domains actually sit and how the two spin orientations are favored.

Key Contributions

1. Demonstration that a "1D" chain in a 2D Rydberg array is not a uniform 1D Ising model. With nearest-neighbor spacing held fixed at 6 μm and net magnetization tuned to about half, the site-resolved magnetization and domain-wall densities remain strongly non-uniform in both geometries.

2. Identification of geometry-induced domain-wall pinning. In the closed 33-atom triangle the domain walls concentrate at the three corners, where the next-nearest-neighbor coupling is stronger by a factor of about 2.4; in the 47-atom square outline walls are pushed away from the corners, whose atoms preferentially occupy the Rydberg state.

3. Evidence of ℤ₂ symmetry breaking from an engineered vacancy. The two atoms flanking the missing atom in the square outline prefer the Rydberg state even though they interact antiferromagnetically with next-nearest-neighbor strength, effectively producing a ferromagnetic bond across the 12 μm gap and suppressing odd numbers of domain walls.

4. Quantitative characterization of domain-wall statistics and wall-type asymmetry. Both datasets are captured by parity-biased binomial fits (p_triangle ≈ 0.14; p_square ≈ 0.16 with parity bias r ≈ 0.19), and among single-wall triangle samples, walls of the form |gg⟩ outnumber |rr⟩ by roughly 3:1.

Main Findings

• Corners pin walls in the triangle. With no atom sitting at the vertices, the three corners have a next-nearest-neighbor separation of √3·a ≈ 10.39 μm rather than the straight-edge 12 μm, giving V_NNN(corner) = V_NN/27 ≈ 4.30 rad/μs versus V_NNN(edge) = V_NN/64 ≈ 1.82 rad/μs — a factor of 64/27 ≈ 2.4 stronger. The energy splitting V_NNN(corner) − V_NNN(edge) ≈ 2.5 rad/μs is enough to lift the translational degeneracy and localize domain walls. This shows up as robust pinning in the wall-density profile and only a faint alternation of average site magnetization.

• The square outline breaks ℤ₂ far more strongly. Because an atom sits at each of the four 90° corners (next-nearest-neighbor distance a√2 ≈ 8.49 μm, V_NNN ≈ V_NN/8 ≈ 14.5 rad/μs), the corner atoms preferentially occupy the Rydberg state, yielding a visible site-resolved antiferromagnetic texture rather than a near-zero average magnetization.

• The vacancy acts as a ferromagnetic bond. The two atoms flanking the missing atom weakly interact antiferromagnetically at next-nearest-neighbor strength, which should favor opposite states and thus an odd number of domain walls. The data instead show a suppression of odd wall counts (bias r ≈ 0.19) and a preference for both atoms to be in the Rydberg state, which the authors attribute to the absence of the Rydberg blockade at the vacancy creating an energetic preference for near-vacancy Rydberg atoms.

• Parity-biased binomial statistics describe both systems. For the closed 33-atom ring, only an odd number of walls is possible, and a parity-biased binomial with p_triangle ≈ 0.14 fits the data well by Kullback–Leibler minimization. For the open 47-atom square, both parities are allowed but odd counts are suppressed, requiring the additional bias parameter r ≈ 0.19 alongside p_square ≈ 0.16.

• |gg⟩ walls dominate over |rr⟩ walls by roughly 3:1. For the 33-atom periodic triangle with a single wall, the |gg⟩ configuration has (N−1)/2 = 16 Rydberg excitations with no adjacent pair, while the |rr⟩ configuration has (N+1)/2 = 17 with one adjacent pair. The classical estimate E↑↑ − E↓↓ ≈ −Δ + V_NN gives about −9 rad/μs at Δ_final = 125 rad/μs and V_NN ≈ 116 rad/μs — so the two wall types are nearly degenerate, with |rr⟩ marginally lower in energy, yet the experiment strongly favors |gg⟩. The authors attribute this to the constrained dynamics enforced during the quasi-adiabatic ramp rather than to the final energy landscape.

• Hardware blockade ratios are large but finite. V_NN/Ω ≈ 116/15.8 ≈ 7.4 for the triangle and ≈ 116/13.0 ≈ 8.9 for the square, so the PXP description is a good but not exact approximation.

• Magnetization calibration succeeded. After post-selection, the triangle run yielded 2,770,350 qubit measurements with 50.867% in the Rydberg state; the square run yielded 3,119,578 qubit measurements with 50.790% in the Rydberg state — close to the even balance an antiferromagnetic Ising simulation would want.

• Ground-state sampling rates are low. Frequent production of multiple domain walls is consistent with the finite 4 μs ramp at the given hardware energy scale producing non-adiabatic defects, possibly compounded by analog control errors or biased qubit decoherence.

• Post-selection losses are substantial. Of 100,000 shots per geometry, 83,950 were retained for the triangle (the rest discarded for atom-filling failure, a roughly 0.007 but geometry-dependent probability) and 66,374 for the square. Of the retained triangle samples, 8,659 had a single domain wall, corresponding to the ground state of the closed odd-spin antiferromagnet.

Methodology in Plain English

The team programmed atoms onto QuEra's Aquila processor in two shapes: a closed triangle with 33 atoms and a square outline with 47 atoms and one atom deliberately removed from the top edge. In both cases neighboring atoms sit 6 μm apart, chosen (above the 4 μm machine minimum) so the Rydberg blockade does not dominate.

Each atom is a two-level system — ground state treated as spin down, Rydberg state as spin up — coupled by van der Waals interactions falling off as 1/r⁶ with C₆ = 5,420,503 μm⁶ rad/μs. The researchers ran a "quasi-adiabatic" schedule: hold the Rabi frequency Ω at a large value while sweeping the detuning Δ from negative to positive over the longest time Aquila allows, 4 microseconds. Ω must ramp up from zero at t = 0 and back to zero between 3.9 and 4 μs. For the triangle, Δ sweeps from −125 to 125 rad/μs with Ω = 15.8 rad/μs (the hardware maximum); for the square, Δ sweeps from −114 to 114 rad/μs with Ω = 13.0 rad/μs.

Before taking data, they independently tuned each schedule so that roughly half the measured atoms land in the Rydberg state, matching what antiferromagnetic ordering would predict. They then collected 100,000 shots per geometry, discarded shots where programmed atoms failed to load, and analyzed the distribution of the number of domain walls, the spatial density of walls along each bond, and the average magnetization per site. Schedules were programmed with Bloqade.jl.

Why This Matters

The work establishes a practical caveat for the growing body of Rydberg-atom experiments that claim to realize one-dimensional spin models: the physical layout of atoms in a two-dimensional plane is itself a perturbation, and corners and vacancies can select spin configurations, bias domain-wall parity, and localize defects even when nearest-neighbor spacing is perfectly uniform. It also shows that the Rydberg blockade's intrinsic asymmetry between |rr⟩ and |gg⟩ walls leaves an observable signature in defect statistics, distinct from what a ℤ₂-symmetric Ising model would produce.

Real-world applications:

• Benchmarking and error characterization for analog quantum simulators. Knowing how geometry distorts an intended Hamiltonian lets users separate hardware artifacts from genuine many-body physics.
• Quantum annealing and optimization. Rydberg arrays are used for Maximum Independent Set and related combinatorial problems; understanding how vacancies and boundaries bias the sampled configurations informs how problem graphs should be embedded.
• Frustrated magnetism and lattice gauge studies. Corner-induced coupling enhancement is a design parameter for engineering frustrated or topological ground states on Kagome-like lattices.
• Defect-engineering in quantum materials emulation. Deliberate vacancies could be used as a tool to pin or guide domain walls rather than purely as an error source.

Industry relevance: For hardware vendors and cloud quantum providers, the results quantify how much of a "1D" experiment's output is set by layout rather than by the intended model, which matters for anyone selling analog quantum sampling as a service or validating device specifications.

Future Directions

• Separate the wall-type asymmetry's origin. The roughly 3:1 preference for |gg⟩ over |rr⟩ walls contradicts a near-degenerate (or slightly |rr⟩-favoring) classical energy estimate; disentangling constrained dynamics during the ramp from control error and decoherence would clarify how much is fundamental.
• Test whether the vacancy-induced ferromagnetic bond is tunable. Varying the gap length around the 12 μm double-nearest-neighbor distance, or the local detuning, would establish how robust the effect is and whether it can be engineered deliberately.
• Improve ground-state fidelity. The low single-domain-wall sampling rate (8,659 of 83,950 retained triangle shots) points to non-adiabatic defect creation within the 4 μs limit; longer ramps or optimized schedules on future hardware are the obvious remedy.
• Extend to true circular or otherwise uniform geometries. The authors note a circular arrangement would be preferable because it would exclude non-uniform further-neighbor interactions; whether Aquila's 4 μm square grid and vertical-alignment constraint permit an adequate approximation is an open design question.

Target Audience

Researchers using neutral-atom analog quantum computers for spin-model emulation, quantum annealing, or combinatorial optimization; condensed-matter physicists interested in domain-wall and defect physics in constrained systems; and quantum-hardware teams who need to characterize how device geometry and atom loading fidelity affect the Hamiltonians their machines actually realize. The paper is most useful to readers with some background in the transverse-field Ising model and the Rydberg blockade, though the core message about geometry-induced pinning and ℤ₂ breaking requires no specialist mathematics.

Authors’ abstract

We perform experiments on QuEra's neutral-Rydberg-atom Aquila quantum computer, using quasi-adiabatic evolution on 1-dimensional models. We use hardware fine-tuning to balance the measurement statistics close to zero net magnetization, in two different geometric outlines: one closed triangular model with 33 atoms, and one 47-atom square with an engineered atom vacancy. The nature of this processor restricts the geometry of the atoms that can be programmed to a 2D plane. We report effects not predicted by the pure 1D Ising model, such as pinning of domain walls and $\mathbb{Z}_2$ symmetry breaking due to the interplay between van der Waals interactions and the Rydberg blockade. In particular, atoms around the atom vacancy and at the vertices of the 1-dimensional square-outline model strongly prefer the Rydberg state, leading to the effective model having a ferromagnetic bond across the vacancy.