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# Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms

## Abstract

Quantum simulation using synthetic systems is a promising route to solve outstanding quantum many-body problems in regimes where other approaches, including numerical ones, fail1. Many platforms are being developed towards this goal, in particular based on trapped ions2,3,4, superconducting circuits5,6,7, neutral atoms8,9,10,11 or molecules12,13. All of these platforms face two key challenges: scaling up the ensemble size while retaining high-quality control over the parameters, and validating the outputs for these large systems. Here we use programmable arrays of individual atoms trapped in optical tweezers, with interactions controlled by laser excitation to Rydberg states11, to implement an iconic many-body problem—the antiferromagnetic two-dimensional transverse-field Ising model. We push this platform to a regime with up to 196 atoms manipulated with high fidelity and probe the antiferromagnetic order by dynamically tuning the parameters of the Hamiltonian. We illustrate the versatility of our platform by exploring various system sizes on two qualitatively different geometries—square and triangular arrays. We obtain good agreement with numerical calculations up to a computationally feasible size (approximately 100 particles). This work demonstrates that our platform can be readily used to address open questions in many-body physics.

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## Data availability

The data presented in the figures and that support the other findings of this study are available from the corresponding author on reasonable request.

## Code availability

The codes are available upon reasonable request from the corresponding author.

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## Acknowledgements

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 817482 (PASQuanS). M.S. acknowledges support by the Austrian Science Fund (FWF) through grant number P 31701 (ULMAC). D.B. acknowledges support from the Ramón y Cajal programme (RYC2018-025348-I). K.-N.S. acknowledges support by the Studienstiftung des Deutschen Volkes. A.A.E. and A.M.L. acknowledge support by the Austrian Science Fund (FWF) through grant number I 4548. The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC) and the LEO HPC infrastructure of the University of Innsbruck.

## Author information

Authors

### Contributions

P.S., M.S., H.J.W. and A.A.E. contributed equally to this work. P.S., H.J.W., D.B., K.-N.S. and V.L. carried out the experiments. M.S., A.A.E., L.-P.H. and T.C.L. performed the simulations. All authors contributed to the data analysis, progression of the project, and on both the experimental and theoretical side. All authors contributed to the writing of the manuscript.

### Corresponding author

Correspondence to Pascal Scholl.

## Ethics declarations

### Competing interests

A.B. and T.C.L. are co-founders and shareholders of Pasqal.

Peer review information Nature thanks the anonymous reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 MPS sampling and scaling with bond dimension χ.

a, b, Average Rydberg density (a) and nearest-neighbour correlation function (b) during the MPS state dynamics on the 10 × 10 square lattice. The black lines show the observables computed from standard tensor contraction and the blue dots show the corresponding sample average of 1,000 generated snapshots. cf, Scaling with bond dimension. c, d, Rydberg density (c) and order parameter mstag (d) during the MPS state dynamics, including experimental imperfections, for different χ on the 10 × 10 square lattice. The insets show the distribution due to the multiple Uij disorder realizations, at final toff = 6 μs for χ = 256. e, f, Scaling of n (e) and mstag (f) with χ at the end of the state preparation protocol for different system sizes. The lightly coloured lines show the multiple disorder instances.

### Extended Data Fig. 2 Testing the coherence of the Rydberg excitation on a single atom.

a, Rabi oscillations showing the probability of measuring the atom in $$|\downarrow \rangle$$ as a function of the excitation time. The line is a fit to the data by the function AeΓtcos(Ωt) + B, yielding Γ = 0.04(1) μs−1, Ω = 2π × 1.32(1) MHz, A = 0.488(3) and B = 0.507(3). b, High-resolution measurement of the first period of the oscillation. Error bars are statistical and often smaller than marker size.

### Extended Data Fig. 3 Benchmarking multiple sweeps on the 4 × 4 array.

ac, Time evolution of sweep shape (a), Rydberg density (b) and staggered magnetization (c) for three distinct sweeps (of durations 2.5 μs (left), 4 μs (middle) and 8 μs (right)) on the 4 × 4 array. In b, c, experimental data are shown in purple circles and the green (red) dashed lines show solutions to the Schrödinger equation (Lindblad master equation, equation (3)). The solid grey lines show solutions of the Schrödinger equation for several random instances of the interaction disorder (see text), the black line is the average over these instances.

### Extended Data Fig. 4 Experimental imperfections.

a, b, Sweep shape for the average Rabi frequency Ω (a) and the average detuning δ (b) versus time. The dashed line shows the proposed protocol and the solid line shows the experimentally obtained parameters. c, d, Spatial dependence of Ω (c) and δ (d) at the maximal values during the protocol. e, f, Distribution of the Rydberg interactions Uij caused by the fluctuations in the atom positions. e, The long-range interactions up to a distance of about 50 μm. f, The distribution of the nearest-neighbour interactions. The dashed vertical line shows the average nearest-neighbour interaction Unnb. The vertical grey line shows, as a reference, the programmed value for non-fluctuating atoms Uprogrammed. g, The left (right) column shows the average density (order parameter mstag) during the sweep for different sizes of the square lattice. Different lines show successive additions of imperfections on the MPS simulations. Starting from the programmed pulse shape without imperfections (blue), we include the real pulse shape measured in the experiment (yellow), add the inhomogeneous fields (green), apply the detection deficiency (red) and, finally, include the interaction disorder from fluctuations in the atom positions. The grey lines show individual samples of atom positions and the black line shows the sample average. Experimental data are shown by circles.

### Extended Data Fig. 5 Effect of vacancies on AF ordering.

a, Histogram of the number of defects for the 14 × 14 array. Out of the roughly 17,000 experimental realizations shown here, we kept only the approximately 500 defect-free shots for the results presented in the main text. b, mstag for different filling fractions. We observe a substantial increase in mstag for defect-free array experiments, compared with approximately 99%-filled-array experiments.

### Extended Data Fig. 6 Long-term stability of the growth of AF ordering on the 8 × 8 array.

Staggered magnetization mstag at different times toff during the sweep shown in Extended Data Fig. 2a with several measurements using the same parameters, realized over 15 h. We observe a dispersion of the measurements due to long-term drift of the experimental setup. The dashed line is a phenomenological fit to the data. The standard error on the mean is smaller than symbol size.

### Extended Data Fig. 7 Growth of AF ordering on a 10 × 10 array during the sweep.

Maps of the connected correlations Ck,l and histograms of the staggered magnetization for different times toff, defined in Fig. 2. The upper (lower) part of the plots show experimental (MPS) results.

### Extended Data Fig. 8 Hypothetical temperature.

a, Assigning a hypothetical temperature. Classical energy density for the instantaneous state $$|{\Psi }({t}_{{\rm{off}}})\rangle$$ of the experiment during the last part of the state preparation protocol (left), and for the corresponding classical equilibrium system versus temperature T (right). Here δ = δf for all datasets. We assign a hypothetical temperature Thyp at each time toff by matching the classical energy, as illustrated by the red line. In the right panel, Tc denotes the critical temperature in the thermodynamic limit N → ∞. b, The Binder cumulant U2 and its (L, 2L) crossing points (black markers) for different linear system dimensions L, which allow the estimation of the critical temperature in the thermodynamic limit. The solid (dotted) grey lines indicate the finite size extrapolated Tc and its standard error. cf, Evolution of the hypothetical temperature. c, Programmed state preparation protocol. df, Corresponding hypothetical temperatures Thyp(toff) during the sweep for different system sizes for the experiment (d) and MPS simulations without (e) and with (f) experimental imperfections. The dashed lines show the classical critical temperature Tc for an infinite system with disorder averaged U/ħ = 2π × 1.86 MHz for δf = 2π × 2 MHz.

### Extended Data Fig. 9 Quantum real-time evolution versus classical equilibrium.

Distribution of the largest Néel cluster sizes smax during (left) and at the end (right) of the quantum time evolution (blue) compared with the classical equilibrium results (yellow) on a 10 × 10 square lattice. a, Experimental results. b, MPS simulation in the setup without including any experimental imperfections. c, MPS simulation including the known experimental imperfections.

### Extended Data Fig. 10 Triangular geometries.

a, b, mstag histogram in the 2/3 plateau obtained from Monte Carlo results on a 108-site triangular cluster with ħδ/U = 4 and temperature T/U = 0.1. a, The real space Rydberg density ni shows that the outermost shell becomes fully populated at low temperature, as also illustrated in the inset, which shows the Rydberg density at the edge. b, The corresponding sublattice magnetization histogram does not reach its full potential width (outer hexagon), as the edge sites cannot participate in the formation of the 2/3-filling states. The dashed hexagon shows the maximum extent of the histogram when only the sites of the system without the edge are considered. c, d, Quantum real-time evolution versus classical equilibrium on the triangular lattice. We plot the distribution of the triangular order parameter mstag at the end of the state preparation protocols entering the 1/3 (c) and 2/3 (d) regimes on a 108-site triangular cluster. Blue (yellow) bars show experimental (corresponding classical) results. The dashed line in d shows the maximal value of mstag in the 2/3 regime induced by the cluster boundaries. e, Distribution of mstag for a 10 × 10 square lattice at the end of the sweep entering the AF phase, as a comparison.

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Scholl, P., Schuler, M., Williams, H.J. et al. Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms. Nature 595, 233–238 (2021). https://doi.org/10.1038/s41586-021-03585-1

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• ### Quantum science with optical tweezer arrays of ultracold atoms and molecules

• Kang-Kuen Ni

Nature Physics (2021)