Quantum Phases of Matter on a 256-Atom Programmable Quantum Simulator

Sepehr Ebadi1, Tout T. Wang1, Harry Levine1, Alexander Keesling1, Giulia Semeghini1, Ahmed Omran1,2, Dolev Bluvstein1, Rhine Samajdar1, Hannes Pichler3,4, Wen Wei Ho1,5, Soonwon Choi6, Subir Sachdev1, Markus Greiner1, Vladan Vuletić7, and Mikhail D. Lukin1 1Department of Physics, Harvard University, Cambridge, MA 02138, USA 2QuEra Computing Inc., Boston, MA 02135, USA 3Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria 4Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020 5 Department of Physics, Stanford University, Stanford, CA 94305, USA. 6Department of Physics, University of California Berkeley, Berkeley, CA 94720, USA 7Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. (Dated: December 24, 2020) Motivated by far-reaching applications ranging from quantum simulations of complex processes in physics and chemistry to quantum information processing [1], a broad effort is currently under- way to build large-scale programmable quantum systems. Such systems provide unique insights into strongly correlated quantum matter [2–6], while at the same time enabling new methods for computation [7–10] and metrology [11]. Here, we demonstrate a programmable quantum simulator based on deterministically prepared two-dimensional arrays of neutral atoms, featuring strong interactions controlled via coherent atomic excitation into Rydberg states [12]. 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 creating and characterizing high-fidelity antiferromagnetically ordered states, and demonstrate the universal properties of an Ising quantum phase transition in (2+1) dimensions [13]. We then create and study several new quantum phases that arise from the interplay between interactions and coherent laser excitation [14], 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.

Recent breakthroughs have demonstrated the poten- tunable interactions, demonstrating several novel quan- tial of programmable quantum systems, with system sizes tum phases and quantitatively probing the associated reaching around fifty trapped ions [2, 15, 16] or super- phase transitions. conducting qubits [7–9], for simulations and computation. Correlation measurements with over seventy pho- tons have been used to perform boson sampling [10], PROGRAMMABLE RYDBERG ARRAYS IN 2D while optical lattices with hundreds of atoms are being used to explore Hubbard models [3–5]. Larger-scale Ising Our experiments are carried out on the second genera- spin systems have been realized using superconducting tion of an experimental platform described previously [6]. elements [17], but they lack the coherence essential for The new apparatus uses a spatial light modulator (SLM) probing quantum matter. to form a large, two-dimensional (2D) array of optical Neutral atom arrays have recently emerged as a tweezers in a vacuum cell (Fig.1a, Methods). This static promising platform for realizing programmable quan- tweezer array is loaded with individual 87Rb atoms from tum systems [6, 12, 18]. Based on individually trapped a magneto-optical trap (MOT), with a uniform loading arXiv:2012.12281v1 [quant-ph] 22 Dec 2020 and detected cold atoms in optical tweezers with strong probability of 50–60% across up to 1000 tweezers. We interactions between Rydberg states [19], atom arrays rearrange the initially loaded atoms into programmable, have been utilized to explore quantum dynamics in one- defect-free patterns using a second set of moving optical and two-dimensional systems [6, 20–24], to create high- tweezers that are steered by a pair of crossed acousto- fidelity [25] and large-scale [26] entanglement, to perform optical deflectors (AODs) to arbitrary positions in 2D parallel quantum logic operations [27, 28], and to realize (Fig.1a) [35]. Our parallel rearrangement protocol (see optical atomic clocks [29, 30]. While large numbers of Methods) enables rearrangement into a wide variety of atoms have been trapped [30] and rearranged in two and geometries including square, honeycomb, and triangular three dimensions [31–34], coherent manipulation of pro- lattices (left panels in Fig. 1b-d). The procedure takes a grammable, strongly interacting systems with more than total time of 50–100 ms for arrays of up to a few hundred a hundred individual particles remains an outstanding atoms and results in filling fractions exceeding 99%. challenge. Here, we realize a programmable quantum Qubits are encoded in the electronic ground state g simulator using arrays of up to 256 neutral atoms with and the highly-excited n = 70 Rydberg state r of each| i | i 2

FIG. 1. Programmable two-dimensional arrays of strongly-interacting Rydberg atoms. a. Atoms are loaded into a 2D array of optical tweezer traps and rearranged into defect-free patterns by a second set of moving tweezers. Lasers at 420 nm and 1013 nm drive a coherent two-photon transition in each atom between ground state |gi = |5S1/2,F = 2, mF = −2i and Rydberg state |ri = |70S1/2, mj = −1/2, mI = −3/2i. b. Fluorescence image of initial random loading of atoms, followed by rearrangement to a defect-free 15×15 (225 atoms) square array. After this initialization, the atoms evolve coherently under laser excitation with Rabi frequency Ω(t) and detuning ∆(t), and long-range interactions Vij . Finally, the state of each atom is read out, with atoms excited to |ri detected as loss and marked with red circles. Shown on the far right is an example measurement following quasi-adiabatic evolution into the checkerboard phase. c, d. Similar evolution on honeycomb and triangular lattices result in analogous ordered phases of Rydberg excitations with ﬁlling 1/2 and 1/3, respectively. atom. We illuminate the entire array from opposite sides CHECKERBOARD PHASE with two counter-propagating laser beams at 420 and 1013 nm, shaped into light sheets (see Methods), to co- The smallest value of R /a that results in an ordered herently couple g to r via a two-photon transition b phase for the quantum many-body ground state of the (Fig.1a). | i | i system corresponds to R /a 1, where only one out of The resulting many-body dynamics U(t) are gov- b every pair of nearest-neighbor≈ atoms can be excited to erned by a combination of the laser excitation and long- r . On a square array, this constraint leads to a Z2- range van der Waals interactions between Rydberg states | i 6 symmetry-broken checkerboard phase with an antiferro- (Vij = V0/ xi xj ), described by the Hamiltonian | − | magnetic (AF) ground state. To realize such a state, H Ω X X X we initialize the array at Rb/a = 1.15 (a = 6.7 µm, = ( gi ri + ri gi ) ∆ ni + Vijninj Ω = 2π 4.3 MHz) with all atoms in g . We then ~ 2 | ih | | ih | − i i i

board ordering as a function of system size (Fig.2e). We a 20 4 find empirically that the probability scales with the num- 10 ber of atoms according to an exponential 0.97N , offer- 0 2 ing a benchmark that includes all experimental imperfec- -10 tions such as finite detection fidelity, non-adiabatic state = 2 ¼ ( M H z ) ¢ = 2 ¼ ( M H z ) -20 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 preparation, spontaneous emission, and residual quan- Time (¹s) tum fluctuations in the ordered state (see Methods). Re- G (2)(k; l) markably, even for a system size as large as 15 15 (225 b -0.25 0.25 c × 0.25 atoms), we still observe the perfect antiferromagnetic 10 0 ) j »H = 11.1(1) +5 ground state with probability 0.10−4% within the expo- ( k ; 225 68 5 ( 2 ) nentially large Hilbert space of dimension 2 10 .

j G ≈ 0.00 l 0 0.25 0 5 10 l ) j »V = 11.3(1)

-5 ( 0 ; (2+1)D ISING QUANTUM PHASE TRANSITION ( 2 ) j G -10 0.00 -10 -5 0 5 10 0 5 10 We now describe quantitative studies of the quantum k Distance phase transition into the checkerboard phase. Quantum d e phase transitions fall into universality classes character- 4 0 10 10 ized by critical exponents that determine universal be- 0:97N 3 1 havior near the quantum critical point, independent of 10 AF AF 10¡ | 1 | 2 the microscopic details of the Hamiltonian [13]. The tran- 2 10 2 10¡ sition into the checkerboard phase is expected to be in the paradigmatic—but never previously observed—quantum 101 3 10¡ Number of States Ising universality class in (2+1) dimensions [14] (with ex- 0

10 Ground State Probability pected dynamical critical exponent z = 1 and correlation 0 20 40 81 121 169 225 length critical exponent ν = 0.629). Number of Occurences Array Size To explore universal scaling across this phase transition for a large system, we study the dynamical build- FIG. 2. Benchmarking of quantum simulator using up of correlations associated with the quantum Kibble- checkerboard ordering. a. A quasi-adiabatic detuning Zurek mechanism [24, 38] on a 16 16 (256 atoms) array, sweep ∆(t) at constant Rabi frequency Ω is used to prepare at fixed R /a = 1.15. We start× at a large negative de- the checkerboard ground state with high fidelity. b. Two- b (2) tuning with all atoms in g and linearly increase ∆/Ω, site correlation function G (k, l), averaged over all pairs of | i atoms on a 12×12 array, showing near-perfect alternating cor- stopping at various points to measure the growth of correlations throughout the entire system. c. Exponential fits of relations across the phase transition (Fig.3a, b). Slower rectified horizontal and vertical correlations are used to ex- sweep rates s = d∆/dt result in longer correlation lengths tract correlation lengths in the corresponding directions ξH ξ, as expected (Fig.3c). and ξV . d. Histogram of many-body state occurrence fre- The quantum Kibble-Zurek mechanism predicts a uni- quency after 6767 repetitions of the experiment on a 12×12 versal scaling relationship between the control parameter array. The two most frequently occurring microstates cor- ξ respond to the two perfect checkerboard orderings, and the ∆ and the correlation length . Specifically, when both next four most common ones are those with a single defect in ∆ and ξ are rescaled with the sweep rate s (relative to a one of the corners of the array. e. Probability of finding a reference rate s0) perfect checkerboard ground state as a function of array size. µ The slightly higher probabilities in odd×odd systems is due to ξ˜ = ξ(s/s0) (2) commensurate edges on opposing sides of the array. All data κ ∆˜ = (∆ ∆c)(s/s0) (3) in this figure are conditioned on defect-free rearrangement of − the array. with exponents µ ν/(1+zν) and κ 1/(1+zν), then universality implies≡ that the rescaled≡ξ˜ −vs. ∆˜ collapses onto a single curve [24] for any sweep rate s. Taking systems [20, 21] by nearly an order of magnitude. z = 1 to be fixed (as expected for a Lorentz-invariant Single-site readout also allows us to study individual theory), we extract ν for our system by finding the value many-body states of our system (Fig.2d). Out of 6767 that optimizes this universal collapse. repetitions on a 12x12 array, the two perfectly ordered In order to obtain ν, we first independently determine states AF1 and AF2 are by far the most frequently the position of the critical point ∆c, which corresponds observed| microstates,i | withi near-equal probabilities be- to the peak of the susceptibility χ = ∂2 H /∂∆2 and tween the two. We benchmark our state preparation by is associated with a vanishing gap [13−]. Forh i adiabiatic measuring the probability of observing perfect checker- evolution under the Hamiltonian (1), the susceptibility χ 4 is related to the mean Rydberg excitation density n by ) a 20 ) z h i z χ ∂ n /∂ 4 = ∆ according to the Hellman-Feynman theo- H h i 10 H M M ( rem. We measure n vs. ∆ along a slow linear sweep to (

0 2 ¼

h i ¼ 2 remain as adiabatic as possible. We take the numerical -10 2 = 0 =

-20 derivative of the fitted data to obtain χ, finding its peak ¢ 0.0 0.5 1.0 1.5 2.0 to be at ∆c/Ω = 1.12(4) (see Methods). Time (¹s) Having identified the position of the critical point, we b ¢==-1 ¢==0 ¢==1 ¢==2 ¢==3 )

ν 2

now extract the value of that optimizes data collapse (

(inset of Fig.3d and Methods). The resulting ν = 0.62(4) G rescales the experimental data to clearly fall on a single c universal curve (Fig.3d). This measurement is in good 5 Sweep Rate s (MHz/¹s) agreement with the predicted ν = 0.629 for the quan- 4 120 15 tum Ising universality class in (2+1) dimensions[14], and 3 distinct from both the mean-field value[13] of ν = 1/2 » and the previously verified value in (1+1) dimensions 2 [24] of ν = 1. Despite imperfections associated with 1 non-adiabatic state preparation and decoherence in our 0 system, this demonstration of universal scaling highlights 0 1 2 3 ¢= opportunities for quantitative studies of quantum critical d phenomena on our platform. º = 0.62(4) 3 D 2 ~ PHASE DIAGRAM OF THE SQUARE LATTICE » 0.5 º 1.0 1 A rich variety of new phases have been recently 0 predicted for the square lattice when Rydberg block- -2 -1 0 1 2 3 4 ade is extended beyond nearest neighbors [14]. To ¢~ map this phase diagram experimentally, we use the Fourier transform of single-shot measurement outcomes P FIG. 3. Observation of the (2+1)D Ising quantum (k) = exp(ik xi/a)ni/√N , which characterizes F i · phase transition on a 16×16 array. a. The transition long-range order in our system. For instance, the checker- into the checkerboard phase is explored using a linear detun- board phase shows a prominent peak at k = (π, π), ing sweep ∆(t) at constant Ω. The resulting checkerboard corresponding to the canonical antiferromagnetic or- ordering is measured at various endpoints. b. Example of (2) der parameter: the staggered magnetization (Fig.4a). growing correlations G with increasing ∆/Ω along a linear We construct order parameters for all observed phases sweep with sweep rate s = 15 MHz/µs. c. Growth of correlation length ξ for s spanning an order of magnitude from using the symmetrized Fourier transform ˜(k1, k2) = F 15 MHz/µs to 120 MHz/µs. ξ used here measures correlations (k1, k2) + (k2, k1) /2, averaged over experimental hF F i between the coarse-grained local staggered magnetization (see repetitions, which takes into account the reflection sym- Methods). d. For an optimized value of the critical expo- metry in our system (see Methods). nent ν, all curves collapse onto a single universal curve when When interaction strengths are increased such that rescaled relative to the quantum critical point ∆c. Inset: dis- next-nearest (diagonal) neighbor excitations are sup- tance D between all pairs of rescaled curves as a function of pressed by Rydberg interactions (R /a √2), trans- ν (see Methods). The minimum at ν = 0.62(4) (red dashed b & line) yields the experimental value for the critical exponent lational symmetry along the diagonal directions is also (red and gray shaded regions indicate uncertainties). broken, leading to the appearance of a new striated phase (Fig.4b). In this phase, Rydberg excitations are mostly located two sites apart and hence appear both on alternating rows and alternating columns. This ordering is ular direction. There are two possible orientations for the immediately apparent through the observation of promi- ordering of this phase, so Fourier peaks are observed at k nent peaks at k = (0, π), (π, 0), and (π, π) in the Fourier = (π, 0) and (π/2, π), as well as at their symmetric part- domain. As discussed and demonstrated below, quantum ners (0, π) and (π, π/2) (Fig.4c). In the thermodynamic fluctuations, appearing as defects on single shot images limit, the star ordering corresponds to the lowest-energy (Fig.4b), play a key role in stabilizing this phase. classical configuration of Rydberg excitations on a square array with a density of 1/4. At even larger values of Rb/a & 1.7, the star phase emerges, with Rydberg excitations placed every four sites We now systematically explore the phase diagram on along one direction and every two sites in the perpendic- 13 13 (169 atoms) arrays, with dimensions chosen to be × 5

Checkerboard Striated Star ~ ~ ~ ~ ~ a 0.0 (k) 6.5 d (¼; ¼) (¼; 0) (¼; 0) (¼=2; ¼) (¼; ¼=2) F -0.7 F ¡ F 2.4 0.1F ¡ F 1.9 0.25 F 0.65 2¼ ® 2 Experiment

1.8 ¼

1.6 a = b R 0 1.4 0 ¼ 2¼ (k) b 0.0 F 3.0 1.2 2¼ ® 1 -2 0 2 4 -2 0 2 4 -2 0 2 4 ¢= ¢= ¢= ¼ e ~(¼; ¼) ~(¼; 0) ~(¼; 0) ~(¼=2; ¼) ~(¼; ¼=2) -1.0 F ¡ F 3.2 0.2F ¡ F 2.0 0.2 F 1.3 2 0 0 ¼ 2¼ Numerics (k) 1.8 c 0.0 F 1.7 2¼ ® 1.6 a = b R 1.4 ¼

1.2

0 1 0 ¼ 2¼ -2 0 2 4 -2 0 2 4 -2 0 2 4 ¢= ¢= ¢=

FIG. 4. Phase diagram of the two-dimensional square lattice. a. Example fluorescence image of atoms in the checkerboard phase and the corresponding Fourier transform averaged over many experimental repetitions hF(k)i, highlighting the peak at (π, π) (circled). b. Image of atoms in the striated phase and the corresponding hF(k)i highlighting peaks at (0, π), (π, 0) and (π, π) (circled). c. Image of atoms in the star phase with corresponding Fourier peaks at (π/2, π) and (π, 0) (circled), as well as at symmetric partners (π, π/2) and (π, 0). d. The experimental phase diagram is constructed by measuring order parameters for each of the three phases for different values of the tunable blockade range Rb/a and detuning ∆/Ω. Red markers indicate the numerically calculated phase boundaries (see Methods). e. The order parameters evaluated numerically using DMRG for a 9×9 array (see Methods). simultaneously commensurate with checkerboard, stri- experimental imperfections. We emphasize that the nu- ated, and star orderings (see Methods). For each value merical simulations evaluate the order parameter for the of the blockade range Rb/a, we linearly sweep ∆ (sim- exact ground state of the system at each point, while ilar to Fig.3a but with a ramp-down time of 200 ns), the experiment quasi-adiabatically prepares these states stopping at evenly spaced endpoints to raster the full via a dynamical process. These results establish the po- phase diagram. For every endpoint, we extract the or- tential of programmable quantum simulators with tun- der parameter corresponding to each many-body phase, able, long-range interactions for studying large quantum and plot them separately to show their prominence in many-body systems that are challenging to access with different regions of the phase diagram (Fig.4d). state-of-the-art computational tools [39]. We compare our observations with numerical simulations of the phase diagram using the density-matrix renormalization group (DMRG) on a smaller 9 9 array QUANTUM FLUCTUATIONS IN THE with open boundary conditions (Fig.4e and red× mark- STRIATED PHASE ers in Fig.4d). We find excellent agreement in the extent of the checkerboard phase. For the striated and star We now explore the nature of the striated phase. In phases, we also find good similarity between experiment contrast to the checkerboard and star phases, which can and theory, although due to their larger unit cells and be understood from a dense-packing argument [14], this the existence of many degenerate configurations, these phase has no counterpart in the classical limit (Ω 0) two phases are more sensitive to both edge effects and (see Methods). Striated ordering allows the atoms→ to 6

a b tion from four next-nearest (diagonal) neighbors on the (0,0) (1,1) (0,0) (1,1) (0,0) sublattice, thus allowing the (1,1) atoms to lower jri r j i their energy by forming a coherent superposition between

Áq = ¼=2 ground and Rydberg states (Fig.5b). ¡ We experimentally study quantum ﬂuctuations in this

Áq = ¼=2 phase by observing the response of the system to short quenches (with quench times tq < 1/Ωq). The depen- g g j i j i dence on the detuning ∆q and laser phase φq of the quench contains information about local correlations and c 1 coherence, which allows us to characterize the quantum d = 0 states on the diﬀerent sublattices. The quench resonance 0.8 for each site depends on the state of its nearest and next- 0.6 nearest neighbors. Due to the large diﬀerence between ( d )

P 0.4 the interaction energies on the (0,0) and (1,1) sublat- d = 3 tices, when one sublattice is resonantly driven, the other 0.2 d = 4 is effectively frozen. 0 The nature of the striated phase is revealed using -10 0 10 20 30 nine-particle operators to measure the state of an atom, ¢q=2¼ (MHz) conditioned on its local environment. Specifically, we d e (d) 1 0.3 evaluate the conditional Rydberg density P , defined as the excitation probability of an atom if all nearest 0.2

( 0 ) 0.5 ( 4 ) neighbors are in g , and exactly d next-nearest (diag- P P | i 0.1 onal) neighbors are in r (see Methods). For d = 0, (0)| i 0 0 we observe a dip in P near the bare atom resonance 0 ¼ 2¼ 0 ¼ 2¼ Áq Áq (Fig.5c), corresponding to resonant de-excitation of the (0,0) sublattice. Meanwhile, P (3) and P (4) have two FIG. 5. Probing correlations and coherence in the stri- separate peaks that correspond to resonant excitation ated phase via quench dynamics. a. Unit cell of striated of the (1,1) sublattice with d = 3 and d = 4 next- ordering (dashed box) with (0,0) and (1,1) sublattices out- nearest neighbor excitations, respectively (Fig.5c). Re- lined in red and blue, respectively. The fill shade on each markably, we find that the quench response of both the site reflects the mean Rydberg excitation. b. The variational (0,0) and (1,1) sublattices depends on the phase φq of states for the (0,0) and (1,1) sublattices are illustrated on the the driving field during the quench (Fig.5d,e). The Bloch sphere (see Methods). The black arrow illustrates the (d) measured visibilities, together with a simple mean-field phase φq of Ω during the quench. c. Probability P of an excitation, conditioned on observing no nearest-neighbor ex- model (see Methods), enable the estimation of unknown citations, and zero (red), three (light blue), or four (dark blue) Bloch vector components on the two sublattices, yielding diagonal next-nearest neighbor excitations. P (0) is plotted for σx = 0.82(6), σy = 0.25(2) for the (0,0) sublattice, h i − h i φq = π/2, showing resonant de-excitation of the (0,0) sub- and σx = 0.45(4), σy = 0.09(1) for the (1,1) sublattice near the bare-atom resonance (leftmost vertical line). lattice.h Wei emphasize− thath i accurate characterization re- (3) (4) P and P are plotted for φq = −π/2, showing excitation quires the use of more sophisticated variational wavefunc- peaks for the (1,1) sublattice at interaction shifts correspond- tions (based on e.g. tensor networks) and warrants fur- ing to 3 or 4 diagonal neighbors (two rightmost vertical lines). ther investigation. This approach can also be extended d, e. P (0) and P (4) vary with quench phase φ at their cor- q through techniques such as shadow tomography [40]. responding resonances (∆q/2π = 1.4 and 20.4 MHz, respectively), demonstrating coherence on both the (0,0) and (1,1) sublattices. Solid line fits are used to extract Bloch vector components. OUTLOOK

These experiments demonstrate that two-dimensional lower their energy by partially aligning with the trans- Rydberg atom arrays constitute a powerful platform for verse field, favoring this phase at finite Ω. This can be programmable quantum simulations with hundreds of seen by considering the 2 2 unit cell, within which qubits. We expect that system size, quantum control one site has a large Rydberg× excitation probability (des- fidelity, and degree of programmability can all be in- ignated the (0,0) sublattice) (Fig.5a). Excitations on creased considerably via technical improvements. In par- its nearest-neighbor (0,1) and (1,0) sublattices are sup- ticular, array sizes and rearrangement fidelities, along pressed due to strong Rydberg blockade. The remaining with atomic state readout, are currently limited by colli- atoms on the (1,1) sublattice have no nearest neighbors in sions with background gas particles, and can be improved the Rydberg state and experience a much weaker interac- with an upgraded vacuum system [25] and increased pho- 7 ton collection efficiency. Quantum coherence can be en- METHODS hanced using higher-power Rydberg lasers and by encoding qubits in hyperfine ground states [19, 28]. Tweezers 2D Optical Tweezer Array with different atomic [25, 30, 41] and molecular [42, 43] Our 2D tweezer array is generated by a free-running 810- species can provide additional features and lead to novel nm Ti:Sapphire laser (M Squared, 18-W pump). The applications in both quantum simulations and metrol- laser illuminates a phase-control spatial light modula- ogy. Finally, rapidly switchable local control beams can tor (Hamamatsu X13138-02), which imprints a computer be used to perform universal qubit operations in parallel generated hologram on the wavefront of the laser field. across the system. The phase hologram is calculated using the phase-fixed Our experiments realize several new quantum phases weighted Gerchberg-Saxton (WGS) algorithm [56] to pro- and provide unprecedented insights into quantum phase duce an arbitrary arrangement of tweezer spots after transitions in two-dimensional systems. These studies propagating to the focus of a microscope objective (Mitu- can be extended along several directions, including the toyo: 3.5 mm glass thickness corrected, 50 , NA=0.5). exploration of non-equilibrium entanglement dynamics Using this method, we can create tweezer× arrays with via rapid quenches across quantum phase transitions [44– roughly 1000 individual tweezers (Extended Data Fig.1). 46], the investigation of topological quantum states of When calculating the phase hologram, we improve trap matter on frustrated lattices [47, 48], the simulation of homogeneity by pre-compensating for the variation in lattice gauge theories [49, 50], and the study of broader diffraction efficiency across the tweezer array (roughly classes of spin models using hyperfine encoding [51]. 2 given by sinc ( π (θ /θ )) where θ denotes the de- Quantum information processing can also be explored 2 trap max flection angle from zeroth order). with hardware-efficient methods for multi-qubit operations [28] and protocols for quantum error correction and We also use the phase control of our SLM to cor- fault tolerant control [52]. Finally, our approach is well rect for optical aberrations on tweezers within the suited for efficient implementation of novel algorithms experimentally-used field of view at the plane of atoms for quantum optimization [53, 54] and sampling [55], en- (Extended Data Fig.2). Aberrations reduce the peak in- abling experimental tests of their performance with sys- tensity of focal spots (characterized by the Strehl ratio), tem sizes exceeding several hundred qubits. and correspondingly reduce the light shift of our tweezers on the atoms. By measuring these light shifts as we scan several low-order Zernike polynomials, we quantify and correct for various aberrations in our optical system. Using this method, we compensate for 70 milliwaves of aberrations, observe a total increase of 18% in our trap intensity (Extended Data Fig.2c), and measure a corresponding reduction in the range of trap frequencies (Ex- tended Data Fig.2d). Aberration correction additionally allows us to place tweezers closer together (minimum sep- aration 3 µm) to reach larger blockade ranges Rb/a. Tweezers in the array have waists 900 nm, trap depths of 2π 17 MHz, and radial trap∼ frequencies of 2π 80∼ kHz.× In each experimental cycle, the tweezers ∼are loaded× from a magneto-optical trap (MOT) with uniform loading probabilities of 50–60% after 50–100 ms loading time.

Atom Rearrangement Atoms are rearranged using an additional set of dynamically moving tweezers, which are overlaid on top of the SLM tweezer array. These movable tweezers are generated by a separate 809-nm laser source (DBR from Photodigm and tapered amplifier from MOGLabs), and are steered with a pair of independently-controlled crossed acousto-optic deflectors (AODs) (AA Opto Elec- tronic DTSX-400). Both AODs are driven by an arbitrary waveform which is generated in real time using our home-built waveform generation software and an arbitrary waveform generator (AWG) (M4i.6631-x8 by Spec- 8 trum Instrumentation). Dynamically changing the RF ward, beginning at a row in which we want to pick up frequency allows for continuous steering of beam posi- an atom, and ending below the bottom row of the array. tions, and multi-frequency waveforms allow for multiple In each downward scan, we eject a single atom from each moving tweezers to be created in parallel [57]. column containing excess atoms; we repeat this process While many 2D sorting protocols have been described until all excess atoms are ejected. previously [31, 33, 35, 58, 59], we implement a novel Parallel sorting within columns: After pre-sorting protocol which is designed to leverage parallel movement and ejection, each column has the correct number of of multiple atoms simultaneously. More specifically, atoms to fill all of its target traps by moving atoms we create a row of moving traps which scans upwards up/down within the column. We now proceed to shuffle along the SLM tweezer array to move one atom within the ith-highest loaded atoms to the ith-highest target each column up in parallel. This is accomplished by traps. As the atoms cannot move through each other, scanning a single frequency component on the vertical in a single vertical scan atoms are moved as close as AOD to move from the bottom to the top of the SLM possible to their target locations, reaching their targets array, during which individual frequency components are unless they are blocked by another atom. We repeat turned on and off within the horizontal AOD to create upward/downward scans until all atoms reach their and remove tweezers at the corresponding columns. target locations. This protocol is designed for SLM tweezer arrays in which traps are grouped into columns and rows. While Rearrangement Parameters and Results this does constrain the possible geometries, most lattice When using moving tweezers to pick up and drop off geometries of interest can still be defined on a subset of atoms in the SLM traps, the moving tweezers ramp on/off points along fixed columns and rows. over 15 µs while positioned to overlap with the corresponding SLM trap. The moving tweezers are approxi- Rearrangement Algorithm mately twice as deep as the static traps, and move atoms Here we detail the rearrangement algorithm, which is il- between SLM traps with a speed of 75 µm/ms. Typi- lustrated in Extended Data Fig. 3. It operates on an cal rearrangement protocols take a total of 50-100 ms to underlying rectangular grid of rows and columns, where implement in practice, depending on the size of the tar- the SLM traps correspond to vertices of the grid. We get array and the random initial loading. Alignment of pre-program a set of ‘target traps’ that we aim to fill. the AOD traps onto the SLM array is pre-calibrated by Pre-sorting: We begin by ensuring that each column measuring both trap arrays on a monitor CMOS camera contains a sufficient number of atoms to fill the target and tuning the AOD frequencies to match positions with traps in that column. In each experimental cycle, due to traps from the SLM array. the random loading throughout the array, some columns A single round of rearrangement results in typical may contain excess atoms while other columns may lack filling fractions of 98.5% across all target traps in a sufficient number of atoms. Accordingly, we apply a the system. This∼ is limited primarily by the finite ‘pre-sorting’ procedure in which we move atoms between vacuum-limited lifetime ( 10 s) and the duration of the columns. To fill a deficient column j, we take atoms from rearrangment procedure.∼ To increase filling fractions, whichever side of j has a larger surplus. We identify we perform a second round of rearrangement (having which atoms to take by finding the nearest atoms from skipped ejection in the first round to keep excess atoms the surplus side which are in rows for which column j for the second round). Since the second round of rear- has an empty trap. We then perform parallel horizontal rangement only needs to correct for a small number of sorting to move these atoms into the empty traps of j defects, it requires far fewer moves and can be performed (not all surplus atoms need to be from the same source more quickly, resulting in less background loss. With column). this approach, we achieve filling fractions of 99.2% ∼ If the one-side surplus is insufficient to fill column j, over more than 200 sites, with a total experimental cycle then we move as many surplus atoms as possible from time of 400 ms. this one side and leave j deficient. We then proceed to the next deficient column, and cycle through until Rydberg Laser System all columns have sufficient atoms. In typical randomly Our Rydberg laser system is an upgraded version of a loaded arrays, this process takes a small number of atom previous setup [26]. The 420-nm laser is a frequency- moves compared to the total number of moves needed for doubled Ti:Sapphire laser (M Squared, 15-W pump). We sorting. This specific algorithm can fail to properly dis- stabilize the laser frequency by locking the fundamen- tribute atoms between columns due to lack of available tal to an upgraded ultra-low expansion (ULE) reference atoms, but these failures are rare and do not limit the cavity (notched cylinder design from Stable Laser Sys- experimental capabilities. tems), with finesse = 30, 000 at 840 nm. The 1013-nm Ejection: After pre-sorting, we eject excess atoms in laser source is anF external-cavity diode laser (Toptica parallel by scanning the vertical AOD frequency down- DL Pro), which is locked to the same reference cavity 9

( = 50, 000 at 1013 nm). To suppress high-frequency imaging the top-hat beams we also correct for remaining phaseF noise from this diode laser, we use the transmitted inhomogeneities by updating the input of our opti- light through the cavity, which is filtered by the narrow mization algorithm (Extended Data Fig.4e,f). Due to cavity transmission spectrum (30 kHz linewidth) [60]. aberrations and imperfections of the vacuum windows, This filtered light is used to injection-lock another laser we observe slightly larger intensity variations on the diode, whose output is subsequently amplified to 10 W atoms than expected ( 3% RMS, 10% peak-to-peak). by a fiber amplifier (Azur Light Systems). ∼ ∼ Using beam shaping optics to homogeneously Rydberg Pulses illuminate the atom array with both Rydberg After initializing our atoms in the ground state g , the lasers, we achieve single-photon Rabi frequen- tweezer traps are turned off for a short time (|

for mi can then be defined as an average over experiment (2) 1 P State Detection repetitions Gm (k, l) = ( mimj mi mj ), N(k,l) i,j h i − h ih i At the end of the Rydberg pulse, we detect the state of where the sum is over all pairs of sites i, j separated atoms by whether or not they are recaptured in our op- by a relative lattice distance of x = (k, l) sites and tical tweezers. Atoms in g are recaptured and detected | i normalized by the number of such pairs N(k,l) (Extended with fidelity 99%, limited by the finite temperature of Data Fig.6c). We obtain the correlation length ξ by the atoms and collisions with background gas particles fitting an exponential decay to the radially averaged in the vacuum chamber. (2) Gm (k, l) (Extended Data Fig.6d). The coarse-grained Atoms excited to the Rydberg state are detected as local staggered magnetization mi is defined such that a loss signal due to the repulsive potential of the opti- (2) the corresponding G (k, l) is isotropic (Extended Data cal tweezers on r . However, the finite Rydberg state m Fig.6c), which makes for natural radial averaging. lifetime[63]( 80| µis for 70S ) leads to a probability of 1/2 This radial average captures correlations across the 15% for r∼ atoms to decay to g and be recaptured entire array better than purely horizontal or vertical ∼by the optical| i tweezers. In our previous| i work [26], we correlation lengths ξ and ξ , which are more sensitive increased tweezer trap depths immediately following the H V to edge effects. Rydberg pulse to enhance the loss signal for atoms in r . In 2D, this approach is less effective because atoms which| i drift away from their initial traps can still be recaptured Determination of the Quantum Critical Point in a large 3D trapping structure created by out-of-plane To accurately determine the location of the quantum interference of tweezers. critical point ∆c for the transition into the checkerboard phase, we measure mean Rydberg excitation n Following an approach similar to what has been h i previously demonstrated[27], we increase the Rydberg vs. detuning ∆ for a slow linear sweep with sweep rate detection fidelity using a strong microwave (MW) pulse s = 15 MHz/µs (Extended Data Fig.7a). To smoothen the measured curve, we fit a polynomial for n vs. ∆ to enhance the loss of atoms in r while leaving atoms h i in g unaffected. The MW source| i (Stanford Research and take its numerical derivative to identify the peak of Systems| i SG384) is frequency-tripled to 6.9 GHz and the susceptibility χ as the critical point[13] (Extended amplified to 3 W (Minicircuits, ZVE-3W-183+). The Data Fig.7b). . Small oscillations in n result from the linear sweep MW pulse, containing both 6 9 GHz and harmonics, h i is applied on the atoms using a microwave horn for not being perfectly adiabatic. To minimize the effect of 100 ns. When applying a Rydberg π-pulse immediately this on our fitting, we use the lowest-degree polynomial followed by the MW pulse, we observe loss probabilities (cubic) whose derivative has a peak, and choose a fit of 98.6(4)%. Since this measurement includes both error window in which the reduced chi-squared metric indicates in the π-pulse as well as detection errors, we apply a a good fit. Several fit windows around ∆/Ω = 0 to 2 give second Rydberg π-pulse after the MW pulse, which good cubic fits, and we average results from each of these transfers most of the remaining ground state population windows to obtain ∆c/Ω = 1.12(4). into the Rydberg state. In this configuration, we observe We also numerically extract the critical point for 99.1(4)% loss probability, which is our best estimate for a system with numerically-tractable dimensions of our Rydberg detection fidelity (Extended Data Fig5). 10 10. Using the density-matrix renormalization group × We find that the loss signal is enhanced by the pres- (DMRG) algorithm, we evaluate n as a function of de- h i ence of both MW fundamental and harmonic frequencies. tuning ∆, and then take the derivative to obtain a peak of the susceptibility at ∆c/Ω = 1.18 (Extended Data Coarse-Grained Local Staggered Magnetization Fig.7c,d). To corroborate the validity of our experimen- We define the coarse-grained local staggered magnetiza- tal fitting procedure, we also fit cubic polynomials to the tion for a site i with column and row indices a and b, DMRG data and find that the extracted critical point respectively, as: is close to the exact numerical value (Extended Data Fig.7d). This numerical estimate of the critical point for a+b ( 1) X a 10 10 array is consistent with the experimental result mi = − (ni nj) × Ni − on a larger 16 16 array. Moreover, our experiments on hj,ii × arrays of different sizes show that ∆c/Ω does not vary where j is summed over nearest neighbors of site i and significantly between 12 12, 14 14, and 16 16 arrays × × × Ni is the number of such nearest neighbors (4 in the (Extended Data Fig.8b). bulk, 3 along the edges, or 2 on the corners). The value of mi ranges from 1 to 1, with the extremal values cor- Data Collapse for Universal Scaling responding to the two− possible perfect antiferromagnetic Optimizing the universal collapse of rescaled correlation orderings locally on site i and its nearest neighbors (Ex- length ξ˜ vs. rescaled detuning ∆˜ requires defining a mea- tended Data Fig.6a,b). The two-site correlation function sure of the distance between rescaled curves for different 11

˜(i) ˜ (i) sweep rates si. Given ξj and ∆j , where the index i at different points in the (∆/Ω,Rb/a) phase diagram us- corresponds to sweep rate si and j labels sequential data ing the density-matrix renormalization group (DMRG) points along a given curve, we define a distance [64] algorithm [65, 66], which operates in the space of the so-called matrix product state (MPS) ansätze. While s 2 1 X X X (i0) (i0) originally developed for one-dimensional systems, DMRG D ξ˜ f (i) ˜ . = j ∆j (4) can also be extended to two dimensions by represent- N 0 − i i 6=i j ing the 2D system as a winding 1D lattice [67], albeit with long-range interactions. A major limitation to two- (i) ˜ ˜(i) The function f (∆) is the linear interpolation of ξj dimensional DMRG is that the number of states required ˜ (i) vs. ∆j , while N is the total number of terms in the to faithfully represent the ground-state wavefunction has three nested sums. The sum over j only includes points to be increased exponentially with the width of the sys- that fall within the domain of overlap of all data sets, tem in order to maintain a constant accuracy. For our avoiding the problem of linear interpolation beyond the calculations, we employ a maximum bond dimension of domain of any single data set. Defined in this way, the 1600, which allows us to accurately simulate 10 10 collapse distance D measures all possible permutations square arrays [14]. We also impose open boundary condi-× of how far each rescaled correlation growth curve is from tions in both directions and truncate the van der Waals curves corresponding to other sweep rates. interactions so as to retain up to third-nearest-neighbor Applied to our experimental data, D is a function couplings. The numerical convergence criterion is set by of both the location of the critical point ∆c and the the truncation error, and the system is regarded to be critical exponent ν (Extended Data Fig.8a). Using well-converged to its true ground state once this error −7 the independently measured ∆c/Ω = 1.12(4), we obtain drops below a threshold of 10 . In practice, this was ν = 0.62(4) for optimal data collapse, and illustrate typically found to be achieved after (102) successive O in particular the better collapse for this value than sweeps. for other values of ν (Extended Data Fig.8c-e). The Since the dimensions of the systems studied in Fig- quoted uncertainty is dominated by the corresponding ure4, (13 13 (experimentally) and 9 9 (numerically), × × uncertainty of the extracted ∆c/Ω, rather than by the are both of the form (4n + 1) (4n + 1), the two phase × precision of finding the minimum of D for a given ∆c/Ω. diagrams are expected to be similar. In particular, both Our experiments give consistent values of ∆c/Ω and ν these system sizes are compatible with the commensu- for systems of size 12 12, 14 14, and 16 16 (Extended rate ordering patterns of the crystalline phases observed Data Fig.8b). × × × in this work, and can host all three phases (at the appropriate Rb/a) with the same boundary conditions. Order Parameters for Many-Body Phases Likewise, for extraction of the QCP, we use a 10 10 × We construct order parameters to identify each array as it is the largest numerically accessible square phase using the Fourier transform to quantify the lattice comparable to the 16 16 array used in our study × amplitude of the observed density-wave order- of the quantum phase transition. ing. We define the symmetrized Fourier transform ˜(k1, k2) = (k1, k2) + (k2, k1) /2 to take into Mean-Field Wavefunction for the Striated Phase F hF F i account the C4 rotation symmetry between possible To understand the origin of the striated phase, it is in- ground-state orderings for some phases. For the star structive to start from a simplified model in which we phase, the Fourier amplitude ˜(π, π/2) is a good assume that nearest-neighbor sites are perfectly block- F order parameter because ordering at k = (π, π/2) is aded. Since we always work in a regime where Rb/a > 1, unique to this phase. The striated phase, on the other this model should also capture the essential physics of hand, shares its Fourier peaks at k = (π, 0) and (0, π) the full Rydberg Hamiltonian. with the star phase, and its peak at k = (π, π) with In the classical limit of Ω = 0, the perfect checker- the checkerboard phase; hence, none of these peaks board state has an energy per site of ∆/2 + V (√2a) + alone can serve as an order parameter. We therefore V (2a), with V (x) being the interaction− between sites construct an order parameter for the striated phase to at a distance x, whereas the corresponding energy for be ˜(0, π) ˜(π/2, π), which is nonzero in the striated the star-ordered state is ∆/4 (neglecting interactions F − F phase and zero in both checkerboard and star. Similarly, for x > 2a). Accordingly,− there is a phase transi- the checkerboard shares its k = (π, π) peak with the tion between the checkerboard and star phases when striated phase, so we construct ˜(π, π) ˜(0, π) as an ∆ = 4[V (√2a) + V (2a)]. On the other hand, for the F − F order parameter which is zero in the striated phase and same density of Rydberg excitations, the striated phase nonzero only in checkerboard. has a classical energy per site of ∆/4 + V (2a)/2, which is always greater than that of the− star phase; hence, stri- Numerical Simulations of the 2D Phase Diagram ated ordering never appears in the classical limit. We numerically compute the many-body ground states At finite Ω, however, the striated phase emerges due 12 to a competition between the third-nearest-neighbor in- 5a). teractions and the second-order energy shift upon dressing a ground state atom off-resonantly with the Rydberg To resolve the local response of the system, we use state. We can thus model the ground state of the striated high-order correlators which are extracted from single- phase as a product state, where (approximately) 1/2 of shot site-resolved readout. In particular, we define an / ˆ(d) the atoms are in the ground state, 1 4 of the atoms are operator i on the eight atoms surrounding site i. This in the Rydberg state, and the remaining 1/4 are in the operatorO projects the neighboring atoms into configura- ground state with a weak coherent admixture of the Ryd- tions in which all four nearest atoms are in g and ex- berg state. A general mean-field ansatz for a many-body actly d of the diagonal neighbors are in r . Specifically,| i wavefunction of this form is given by (d) | i the operator ˆ decomposes into a projector Aî on the Oi O four nearest neighboring atoms and Bˆ(d) on the four di- Ψ (a , a ) = (cos a1 g i + sin a1 r i) (5) i | str 1 2 i | i | i (d) (d) i∈A agonal neighbors, according to ˆ = AîBˆ . Defining 1 Oi i O O n¯i = g i g and ni = r i r , the nearest neighbor projec- (cos a2 g i + sin a2 r i) g j, | i h | ˆ | Qi h | | i | i | i tor is written as Ai = hj,ii n¯j, where . denotes near- i∈A2 j∈B h i ˆ(d) est neighbors. The projector Bi sums over all configu- where A and A represent the two sublattices of the (bi- 1 2 rations of the diagonal neighbors (indexed k1, k2, k3, k4) partite) A sublattice, and a1,2 are variational parameters. with d excitations: If a1 = a2, then our trial wavefunction simply represents ˆ(4) a checkerboard state, but if a1 = a2, this state is not of Bi = nk1 nk2 nk3 nk4 (6) the checkerboard type, and leads6 to the striated phase. Bˆ(3) =n ¯ n n n + n n¯ n n + ... (7) Based on this ansatz, we can now explicitly see how i k1 k2 k3 k4 k1 k2 k3 k4 ˆ(2) the striated phase may become energetically favorable Bi =n ¯k1 n¯k2 nk3 nk4 +n ¯k1 nk2 n¯k3 nk4 + ... (8) in the presence of a nonzero Ω. Consider the atoms on the partially excited sublattice to be in the superposition These operators are used to construct the conditional g +[Ω/ 4V (√2a) ∆ ] r ; this describes the state of the Rydberg density | i { , − } | i atoms on the (1 1) sublattice in the notation of Fig.5. P (d) ni ˆ The net energy per site of the system is then P (d) = ih Oi i P ˆ(d) ∆ V (2a) Ω2 Ω2 V (√2a) ihOi i + + − 4 2 − 4 (4V (√2a) ∆) 2 (4V (√2a) ∆)2 which measures the probability of Rydberg excitation on − − site i surrounded by neighboring-atom configurations for where the third and fourth terms are the second-order ˆ(d) which i = 1. energy shift and mean-field interaction shift, respec- To quantifyO coherences, we measure these conditional tively. From this expression, we observe that if the probabilities on their corresponding resonances, after a energy gained from the dressing (these last two terms) fixed quench with variable quench phase φq. For a single is larger than V (2a)/2, then the striated phase prevails particle driven by the Hamiltonian H = Ω(cos φqσx + over the star phase. sin φqσy)/2 + ∆σz/2 for time τ, the resulting Heisenberg evolution is given by σ0 = U †σ U, where U = e−iHτ . Dynamical Probe of the Striated Phase z z The resulting operator can be expressed as We prepare striated ordering using an optimized cubic 0 spline sweep along Rb/a = 1.47, ending at ∆/Ω = 2.35. σ = Ω˜ sin 2α( σx sin φq + σy cos φq) (9) z − Immediately after this sweep, the system is quenched 2 + 2∆˜ Ω˜ sin α(σx cos φq + σy sin φq) (10) to detuning ∆q and relative laser phase φq. We quench 2 2 2 at a lower Rabi frequency Ω = Ω/4 2π 1 MHz to + (cos α (1 2∆˜ ) sin α)σz (11) q − − improve the resolution of this interaction≈ spectroscopy.× For the chosen lattice spacing, the interaction energy where ∆˜ = ∆/√∆2 + Ω2, Ω˜ = Ω/√∆2 + Ω2, and α = 1 √ 2 2 between diagonal excitations is 2π 5.3 MHz. The 2 τ ∆ + Ω . reference phase for the atoms φ =× 0 is set by the We fit the conditional probabilites P (0) and P (4) as instantaneous phase of the Rydberg coupling laser at a function of φq (Fig.5d,e), taking ∆ as the effective the end of the sweep into striated ordering. In the Bloch detuning from interaction-shifted resonance, and mea- 0 sphere picture, φ = 0 corresponds to the +x axis, so the suring σ as a function of φq to extract the Bloch vector h zi wavefunctions on (0,0) and (1,1) sublattices correspond components σx , σy , σz on the two respective sub- to vectors pointing mostly up or mostly down with a lattices. Forh thei (1,1)h i sublatticeh i response, we model the small projection of each along the +x axis. In the same evolution averaged over random detunings, due to 15% ∼ Bloch sphere picture, quenching at φq = π/2 or π/2 fluctuations of the interaction shifts associated with corresponds to rotations around the +y or y axes− (Fig. thermal fluctuations in atomic positions, which broaden − 13 and weaken the spectroscopic response. For both sublattices we also include fluctuations in the calibrated pulse area ( 10% due to low power used). The extracted ∼ (0,0) fit values are σx,y,z = 0.82(6), 0.25(2), 0.32(4), and (1,1) − − σx,y,z = 0.46(4), 0.01(1), 0.91(5). − Acknowledgements We thank many members of the Harvard AMO community, particularly Elana Urbach, Samantha Dakoulas, and John Doyle for their efforts enabling safe and productive operation of our laboratories during 2020. We thank Hannes Bernien, Dirk Englund, Manuel Endres, Nate Gemelke, Donggyu Kim, Peter Stark, and Alexander Zibrov for discussions and experimental help. We acknowledge financial support from the Center for Ultracold Atoms, the National Science Foundation, the Vannevar Bush Faculty Fellowship, the U.S. Department of Energy, the Office of Naval Research, the Army Research Office MURI, and the DARPA ONISQ program. 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 U.S. Department of Energy under Grant DE-SC0019030. The DMRG calculations were performed using the ITensor Library [68]. 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 contributions S.E., T.T.W., H.L., A.K., G.S, A. O. and D.B. contributed to the building experimental setup, performed the measurements, and analyzed 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.

Competing interests M.G., V.V., and M.D.L. are co-founders and shareholders of QuEra Computing. A.O. is a shareholder of QuEra Computing.

Correspondence and requests for materials should be addressed to M.D.L. 14

EXTENDED DATA FIG. 1. Large arrays of optical tweezers. The experimental platform produces optical tweezer arrays with up to ∼ 1000 tweezers and ∼ 50% loading probability per tweezer after 100 ms of MOT loading time. a. Camera image of an array of 34×30 tweezers (1020 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.

a c

b d

EXTENDED DATA FIG. 2. Correcting for aberrations in the SLM tweezer array. The aberration correction procedure utilizes 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 thirteen lowest order Zernike polynomials. The sum of all polynomials with their associated coeﬃcients gives the total aberrated phase proﬁle in the optical system, which is now corrected (total RMS aberration of ∼ 70 milliwaves). 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 0 depths are measured by probing the light shift of each trap on the |5S1/2,F = 2i → |5P3/2,F = 2i 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). 15

a b Row 8 Vertical AOD Vertical AOD Row 1

Column 4 Horizontal AOD Column 1

time Horizontal AOD

Loading 1. 2. 3. 4. 5. 6.

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 downwards move. 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 which 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.

a c Initial Hologram e Hologram Correction

Before Feedback After Feedback b d f

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 this data is used as an input for the hologram optimization algorithm. b. Corrected and measured wavefront error through our optical system, 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 significantly improved homogeneity. 16

a Rydberg laser b Rydberg laser MW source MW source Tweezers Tweezers

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 |gi, exciting to |ri with a Rydberg π-pulse, and then applying the MW 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 |ri = |70Si to other Rydberg states. Note that the transition to |69P i and |70P i are in the range of 10 − 12 GHz, and over this entire range there is strong transfer out of |ri. Other resonances might be due to multi-photon effects. b. With fixed 6.9-GHz MW frequency and varying pulse time, there is a rapid transfer out of the Rydberg state on the timescale of several nanoseconds. Over short time-scales, there may be coherent oscillations which return population back to |ri, so a 100 ns pulse is used for enhancement of loss signal of |ri in the experiment.

a b c 1 1 d j ) i 2 1 ( m m

G » = 3.4 j » = 1.2 j

-1 0 ) 2 ( m

1 1 G j

0.1 j ) i 2 ( m m

G 0 5 10 j Radial Distance -1 0

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) (2) compared to for the faster sweep (top), as expected from the Kibble-Zurek mechanism. c. Isotropic correlation functions Gm for the corresponding coarse-grained local staggered magnetizations after a faster (top) or a slower (bottom). d. As a function (2) of radial distance, correlations Gm decay exponentially with a length scale corresponding to the correlation length ξ. The two decay curves correspond to faster (orange) and slower (blue) sweeps. 17

Experiment: 16x16 Numerics: 10x10 a d 0.5 0.5 ® ® n n

y 0.4 y 0.4 t t i i s s n n

e 0.3 e 0.3 D D

g g

r 0.2 r 0.2 e e b b

d 0.1 d 0.1 y y R R 0 0 -2 -1 0 1 2 3 -2 -1 0 1 2 3 ¢= ¢=

b e

0.25 0.15 Â Â 0.2 y y t t i i l l i i b b

i i 0.15 t t

p 0.1 p

e e 0.1 c c s s u u

S S 0.05 0.05 ¢ = 1:12(4) c ¢c = 1:18 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ¢= ¢=

c f 2.5

4 2

) ) 3

¼ 1.5 ¼ ; ; ¼ ¼

( ( 2

~ 1 ~ F F

1 0.5

0 -2 -1 0 1 2 3 -2 -1 0 1 2 3 ¢= ¢=

EXTENDED DATA FIG. 7. Extracting the quantum critical point. a. The mean Rydberg excitation density hni vs. detuning ∆/Ω on a 16×16 array. The data is fitted within a window (dashed lines) to a cubic polynomial (red curve) as a means of smoothening the data. b. The peak in the numerical derivative of the fitted data (red curve) corresponds to the critical point ∆c/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 (gray) is too noisy to be useful. c. Order parameter F˜(π, π) for the checkerboard phase vs. ∆/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 hni vs. ∆/Ω on a 10×10 array. For comparison against the experimental fitting procedure, the data from numerics is 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 ∆c/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 F˜(π, π) vs. ∆/Ω, with the exact quantum critical point from numerics shown (red line). 18

a 0 10 b 1.0 1.4 8 0.5 1.2

) 0.8 6 . b

r / =

1 a 1.0 c c (

4 ¢ ¢

D 0.6 0.8 1.5 2 0.4 0.6 12 12 14 14 16 16 2 0 £ £ £ 0.4 0.6 0.8 1.0 Array Size º c d e 2.5 º = 0:5 2.5 º = 0:62 2.5 º = 1

2.0 2.0 2.0 ~ ~ ~ » 1.5 » 1.5 » 1.5

1.0 1.0 1.0

0.5 0.5 0.5 -0.5 0.0 0.5 1.0 -0.5 0.0 0.5 1.0 -0.5 0.0 0.5 1.0 ¢~ ¢~ ¢~

EXTENDED DATA FIG. 8. Optimization of data collapse. a. Distance D between rescaled correlation length ξ˜ vs. ∆˜ curves depends on both the location of the quantum critical point location ∆c/Ω and on the correlation length critical exponent ν. The independently determined ∆c/Ω (blue line, with uncertainty range in gray) 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 ∆c/Ω (blue) is consistent across arrays of diﬀerent sizes. c-e. Data collapse is clearly better at the experimentally determined value (ν = 0.62) as compared to 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 data sets. 19

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