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# Quantum phases of matter on a 256-atom programmable quantum simulator

## Abstract

Motivated by far-reaching applications ranging from quantum simulations of complex processes in physics and chemistry to quantum information processing1, a broad effort is currently underway to build large-scale programmable quantum systems. Such systems provide insights into strongly correlated quantum matter2,3,4,5,6, while at the same time enabling new methods for computation7,8,9,10 and metrology11. Here we demonstrate a programmable quantum simulator based on deterministically prepared two-dimensional arrays of neutral atoms, featuring strong interactions controlled by coherent atomic excitation into Rydberg states12. Using this approach, we realize a quantum spin model with tunable interactions for system sizes ranging from 64 to 256 qubits. We benchmark the system by characterizing high-fidelity antiferromagnetically ordered states and demonstrating quantum critical dynamics consistent with an Ising quantum phase transition in (2 + 1) dimensions13. We then create and study several new quantum phases that arise from the interplay between interactions and coherent laser excitation14, experimentally map the phase diagram and investigate the role of quantum fluctuations. Offering a new lens into the study of complex quantum matter, these observations pave the way for investigations of exotic quantum phases, non-equilibrium entanglement dynamics and hardware-efficient realization of quantum algorithms.

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## Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

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## Acknowledgements

We thank many members of the Harvard AMO community, particularly E. Urbach, S. Dakoulas and J. Doyle for enabling safe and productive operation of our laboratories during 2020. We thank H. Bernien, D. Englund, M. Endres, N. Gemelke, D. Kim, P. Stark and A. Zibrov for discussions and experimental help. We acknowledge financial support from the Center for Ultracold Atoms, the US National Science Foundation, the Vannevar Bush Faculty Fellowship, the US Department of Energy (DE-SC0021013 and DOE Quantum Systems Accelerator Center, contract no. 7568717), the Office of Naval Research, the Army Research Office MURI and the DARPA ONISQ programme. T.T.W. acknowledges support from Gordon College. H.L. acknowledges support from the National Defense Science and Engineering Graduate (NDSEG) fellowship. G.S. acknowledges support from a fellowship from the Max Planck/Harvard Research Center for Quantum Optics. D.B. acknowledges support from the NSF Graduate Research Fellowship Program (grant DGE1745303) and the Fannie and John Hertz Foundation. W.W.H. is supported by the Moore Foundation’s EPiQS Initiative grant no. GBMF4306, the NUS development grant AY2019/2020, and the Stanford Institute of Theoretical Physics. S.C. acknowledges support from the Miller Institute for Basic Research in Science. R.S. and S.S. were supported by the US Department of Energy under grant DE-SC0019030. The DMRG calculations were performed using the ITensor Library73. The computations in this paper were run on the FASRC Cannon cluster supported by the FAS Division of Science Research Computing Group at Harvard University.

## Author information

Authors

### Contributions

S.E., T.T.W., H.L., A.K., G.S, A.O. and D.B. contributed to building the experimental set-up, performed the measurements and analysed the data. Theoretical analysis was performed by R.S., H.P., W.W.H. and S.C. All work was supervised by S.S., M.G., V.V. and M.D.L. All authors discussed the results and contributed to the manuscript.

### Corresponding author

Correspondence to Mikhail D. Lukin.

## Ethics declarations

### Competing interests

M.G., V.V. and M.D.L. are co-founders and shareholders of QuEra Computing. A.K. and A.O. are shareholders of QuEra Computing. All other authors declare no competing interests.

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 Large arrays of optical tweezers.

The experimental platform produces optical tweezer arrays with up to ~1,000 tweezers and ~50% loading probability per tweezer after 100 ms of magneto-optical trap loading time. a, Camera image of an array of 34 × 30 tweezers (1,020 traps), including aberration correction. b, Sample image of random loading into this tweezer array, with 543 loaded atoms. Atoms are detected on an EMCCD camera with fluorescence imaging.

### Extended Data Fig. 2 Correcting for aberrations in the SLM tweezer array.

The aberration correction procedure uses the orthogonality of Zernike polynomials and the fact that correcting aberrations increases tweezer light shifts on the atoms. To independently measure and correct each aberration type, Zernike polynomials are added with variable amplitude to the SLM phase hologram, with values optimized to maximize tweezer light shifts. a, Two common aberration types: horizontal coma (upper) and primary spherical (lower), for which ~50 milliwaves compensation on each reduces aberrations and results in higher-depth traps. b, Correcting for aberrations associated with the 13 lowest-order Zernike polynomials. The sum of all polynomials with their associated coefficients gives the total wavefront correction (RMS ~70 milliwaves) for our optical system, which is summed with the optical tweezer hologram on the SLM. c, Trap depths across a 26 × 13 trap array before and after correction for aberrations. Aberration correction results in tighter focusing (higher trap light shift) and improved homogeneity. Trap depths are measured by probing the light shift of each trap on the $$|5{S}_{1/2},F=2\rangle \to |5{P}_{3/2},F{\prime} =2\rangle$$ transition. d, Aberration correction also results in higher and more homogeneous trap frequencies across the array. Trap frequencies are measured by modulating tweezer depths at variable frequencies, resulting in parametric heating and atom loss when the modulation frequency is twice the radial trap frequency. The measurement after correction for aberrations shows a narrower spectrum and higher trap frequencies (averaged over the whole array).

### Extended Data Fig. 3 Rearrangement protocol.

a, Sample sequence of individual rearrangement steps. There are two pre-sorting moves (1, 2). Move 3 is the single ejection move. Moves 4–6 consist of parallel vertical sorting within each column, including both upward and downward moves. The upper panel illustrates the frequency spectrum of the waveform in the vertical and horizontal AODs during these moves, with the underlying grid corresponding to the calibrated frequencies that map to SLM array rows and columns. b, Spectrograms representing the horizontal and vertical AOD waveforms over the duration of a single vertical frequency scan during a realistic rearrangement procedure for a 26 × 13 array. The heat-maps show frequency spectra of the AOD waveforms over small time intervals during the scan.

### Extended Data Fig. 4 Generating homogeneous Rydberg beams.

a, Measured Gaussian-beam illumination on the SLM for shaping the 420-nm Rydberg beam. A Gaussian fit to these data is used as an input for the hologram optimization algorithm. b, Measured wavefront error through our optical system (after correction), showing a reduction of aberrations to λ/100. c, Computer-generated hologram for creating the 420-nm top-hat beam. d, Measured light intensity of the 420-nm top-hat beam (top), and the cross-section along where atoms will be positioned (bottom). Vertical lines denote the 105-μm region where the beam should be flat. e, Using the measured top-hat intensity, a phase correction is calculated for adding to the initial hologram. f, Resulting top-hat beam after feedback shows considerably improved homogeneity. pk–pk, peak to peak.

### Extended Data Fig. 5 Characterizing microwave-enhanced Rydberg detection fidelity.

The effect of strong microwave (MW) pulses on Rydberg atoms is measured by preparing atoms in $$|g\rangle$$, exciting to $$|r\rangle$$ with a Rydberg π-pulse, and then applying the microwave pulse before de-exciting residual Rydberg atoms with a final Rydberg π-pulse. (The entire sequence occurs while tweezers are briefly turned off.) a, Broad resonances are observed with varying microwave frequency, corresponding to transitions from $$|r\rangle$$ = $$|70S\rangle$$ to other Rydberg states. Note that the transitions to $$|69P\rangle$$ and $$|70P\rangle$$ are in the range of 10–12 GHz, and over this entire range there is strong transfer out of  $$|r\rangle$$. Other resonances might be due to multiphoton effects. b, With fixed 6.9-GHz microwave frequency and varying pulse time, there is a rapid transfer out of the Rydberg state on the timescale of several nanoseconds. Over short timescales, there may be coherent oscillations that return population back to $$|r\rangle$$, so a 100-ns pulse is used for enhancement of loss signal of $$|r\rangle$$ in the experiment.

### Extended Data Fig. 6 Coarse-grained local staggered magnetization.

a, Examples of Rydberg populations ni after a faster (top) and slower (bottom) linear sweep. b, Corresponding coarse-grained local staggered magnetizations mi clearly show larger extents of antiferromagnetically ordered domains (dark blue or dark red) for the slower sweep (bottom) than for the faster sweep (top), as expected from the Kibble–Zurek mechanism. c, Isotropic correlation functions $${G}_{m}^{(2)}$$ for the corresponding coarse-grained local staggered magnetizations after a faster (top) or a slower (bottom) sweep. d, As a function of radial distance, correlations $${G}_{m}^{(2)}$$ decay exponentially with a length scale corresponding to the correlation length ξ. The two decay curves correspond to faster (orange) and slower (blue) sweeps.

### Extended Data Fig. 7 Extracting the quantum critical point.

a, The mean Rydberg excitation density $$\langle n\rangle$$ versus detuning Δ/Ω on a 16 × 16 array. The data are fitted within a window (dashed lines) to a cubic polynomial (red curve) as a means of smoothing the data. b, The peak in the numerical derivative of the fitted data (red curve) corresponds to the critical point Δc/Ω = 1.12(4) (red shaded regions show uncertainty ranges, obtained from varying fit windows). In contrast, the point-by-point slope of the data (grey) is too noisy to be useful. c, Order parameter $$\mathop{ {\mathcal F} }\limits^{ \sim }({\rm{\pi }},{\rm{\pi }})$$ for the chequerboard phase versus Δ/Ω measured on a 16 × 16 array with the value of the critical point from b superimposed (red line), showing the clear growth of the order parameter after the critical point. d, DMRG simulations of  $$\langle n\rangle$$ versus Δ/Ω on a 10 × 10 array. For comparison against the experimental fitting procedure, the data from numerics are also fitted to a cubic polynomial within the indicated window (dashed lines). e, The point-by-point slope of the numerical data (blue curve) has a peak at Δc/Ω = 1.18 (blue dashed line), in good agreement with the results (red dashed line) from both the numerical derivative of the cubic fit on the same data (red curve) and the result of the experiment. f, DMRG simulation of $$\mathop{ {\mathcal F} }\limits^{ \sim }({\rm{\pi }},{\rm{\pi }})$$ versus Δ/Ω, with the exact quantum critical point from numerics shown (red line).

### Extended Data Fig. 8 Optimization of data collapse.

a, Distance D between rescaled correlation length $$\mathop{\xi }\limits^{ \sim }$$ versus $$\mathop{\varDelta }\limits^{ \sim }$$ curves depends on both the location of the quantum critical point Δc/Ω and on the correlation length critical exponent ν. The independently determined Δc/Ω (blue line, with uncertainty range in grey) and the experimentally extracted value of ν (dashed red line, with uncertainty range corresponding to the red shaded region) are marked on the plot. b, Our determination of ν (red) from data collapse around the independently determined Δc/Ω (blue) is consistent across arrays of different sizes. ce, Data collapse is clearly better at the experimentally determined value (ν = 0.62) as compared with the mean-field (ν = 0.5) or the (1 + 1)D (ν = 1) values. The horizontal extent of the data corresponds to the region of overlap of all rescaled datasets.

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Ebadi, S., Wang, T.T., Levine, H. et al. Quantum phases of matter on a 256-atom programmable quantum simulator. Nature 595, 227–232 (2021). https://doi.org/10.1038/s41586-021-03582-4

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