Abstract
Spin models are the prime example of simplified manybody Hamiltonians used to model complex, strongly correlated realworld materials^{1}. However, despite the simplified character of such models, their dynamics often cannot be simulated exactly on classical computers when the number of particles exceeds a few tens. For this reason, quantum simulation^{2} of spin Hamiltonians using the tools of atomic and molecular physics has become a very active field over the past years, using ultracold atoms^{3} or molecules^{4} in optical lattices, or trapped ions^{5}. All of these approaches have their own strengths and limitations. Here we report an alternative platform for the study of spin systems, using individual atoms trapped in tunable twodimensional arrays of optical microtraps with arbitrary geometries, where filling fractions range from 60 to 100 per cent. When excited to highenergy Rydberg D states, the atoms undergo strong interactions whose anisotropic character opens the way to simulating exotic matter^{6}. We illustrate the versatility of our system by studying the dynamics of a quantum Isinglike spin1/2 system in a transverse field with up to 30 spins, for a variety of geometries in one and two dimensions, and for a wide range of interaction strengths. For geometries where the anisotropy is expected to have small effects on the dynamics, we find excellent agreement with ab initio simulations of the spin1/2 system, while for strongly anisotropic situations the multilevel structure of the D states has a measurable influence^{7,8}. Our findings establish arrays of single Rydberg atoms as a versatile platform for the study of quantum magnetism.
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Acknowledgements
We thank H. Busche for contributions in the early stages of the experiment, I. Lesanovsky, H. P. Büchler and T. Pohl for discussions, and Y. Sortais for a reading of the manuscript. This work benefited from financial support by the EU (FETOpen Xtrack Project HAIRS, H2020 FETPROACT Project RySQ, and EU MarieCurie Program ITN COHERENCE FP7PEOPLE2010ITN265031 (H.L.)), by the ‘PALM’ Labex (project QUANTICA) and by the Région ÎledeFrance in the framework of DIM NanoK.
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Reviewer Information Nature thanks C. F. Roos and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Extended data figures and tables
Extended Data Figure 1 Full and partial loading of arrays.
a, Sketch of the experimental sequences. During loading, the camera images are analysed continuously to extract the number of loaded traps. As soon as a triggering criterion is met, the loading is stopped and an image of the initial configuration is acquired. After Rydberg excitation, a final image is acquired, revealing the atoms excited to Rydberg states (green disks at bottom right). b, Average triggering time T_{N} when the triggering criterion is set to N = N_{t}: achieving full loading requires an exponentially long time, limiting the method in practice to N_{t} ≤ 9. The triggering times can vary substantially depending on the density of the magnetooptical trap used to load the array, and the data points shown here correspond to typical conditions used for the data of the main text. Error bars, s.e.m. c, Probability p_{N} of having a number N of loaded traps in the partially loaded regime for the 30trap ‘racetrack’ (left) and the 49trap square array (right; blue dots). The shaded distributions correspond to what would be observed with random triggering. For this partial loading regime, the triggering rate of the experiment is about 1 s^{−1}.
Extended Data Figure 2 Effect of detection errors.
a, Experimental determination of ε. From the data of the full blockade experiments (Fig. 2 of main text), we plot the probability P_{0} of recapturing all N atoms for τ = 0. The solid line is a fit to the expected dependence (1 − ε)^{N}, giving ε = 3% (the shaded area corresponds to 2% < ε < 4%). Error bars, s.e.m. b, Calculated probabilities of observing 0,1 or 2 excitations (columns 1–3) as a function of the excitation pulse area Ωt, assuming a perfect blockade and ε = 3%, for atom numbers N = 3, 9, 15 (rows 1–3).
Extended Data Figure 3 Full data set for the Rydberg blockade data.
a, Fully loaded arrays of 1 to 9 traps (n = 82). b, Partially loaded array of N_{t} = 19 traps, containing from N = 10 to N = 15 atoms (n = 100). The column on the left shows the probability P_{0} of recapturing all atoms, the centre column the probability P_{1} of losing just one atom out of N, and the column on the right the probability P_{2} of losing two atoms out of N. The solid lines are fits by equation (5). Error bars, s.e.m. c, Damping rate γ extracted from the P_{0} data as a function of the number of atoms in the array. Error bars, s.d.
Extended Data Figure 4 Homogeneous excitation in the eightatom ring.
a, For Ωτ = 3.1, we observe strongly contrasted oscillations in the pair correlation function g^{2}(k). b, The average density of Rydberg excitations, however, is approximately the same on every site. The horizontal dashed line indicates the mean over all sites. Error bars, s.e.m.
Extended Data Figure 5 Full time evolution of the correlation functions for the 30trap, racetrackshaped chain.
a, Same as for Fig. 3a–c. The right panel is the time evolution of the pair correlation function, clearly showing that, for times longer than a few Ω^{−1}, the pair correlation function does not evolve significantly. The vertical dashed line indicates the value of the blockade radius. b, The principal quantum number is now n = 57, and the Rabi frequency Ω = 2π × 1.7 MHz, such that R_{b} = 2.4a. The central panel shows the time evolution of the Rydberg fraction, and the right panel the time evolution of the pair correlation function. For both a and b, f_{R} approaches, at long times, the closepacking limit a/R_{b} of hard rods of length R_{b} (dashed horizontal lines).
Extended Data Figure 6 Full time evolution of the experimental correlation function for the 7 × 7 square array.
One observes the blockaded region around (k, l) = (0, 0), with a slight flattening reflecting the anisotropy of the interaction. After a few Ω^{−1}, the correlation function g^{(2)}(k, l) does not evolve any more.
Extended Data Figure 7 Full versus partial loading for the dynamics and correlations in the case of Fig. 4a–c.
a, Rydberg fraction as a function of time for the partially loaded (solid line) or fully loaded (thin dashed line) 30trap array. b, Pair correlation function g^{(2)}(k) for Ωτ ≈ 2.0, for the partially loaded (solid line) or fully loaded (thin dashed line) 30trap array. In both cases, the effect of detection errors (ε = 3%) is included.
Extended Data Figure 8 Assessing the validity of the approximation of translational invariance in the eightatom ring.
Calculated pair correlation function g^{(2)}(k) as a function of the excitation time for the eightatom ring. a, Simulation using the experimentally relevant anisotropic interaction, which breaks translational invariance. b, Simulation with the same parameters as in a, except that the angular dependence is neglected (we replace equation (8) by its value for θ = 0), thus reestablishing translational invariance. We observe that the contrast in a is reduced, as expected, but only in a marginal way.
Extended Data Figure 9 Effect of the Zeeman structure of the Rydberg states on the dynamics of the Rydberg fraction f_{R}.
We use the toy model of nP_{1/2} Rydberg states discussed in Methods, with n = 30 and B = 0.2 G. a, The atoms are aligned along the quantization axis z, and spaced by a = 1.6 μm (inset). In this case, the full solution including the Zeeman structure (red solid line) agrees perfectly with the solution of the effective spin1/2 model with an anisotropic effective potential (as used in all the rest of the paper, black dotted line). b, The atoms are aligned perpendicularly to z, with a = 2 μm (inset; thus keeping the same effective potential interaction between adjacent atoms as in a). The full (red solid line) and approximate (black dotted line) solutions agree at short times, but for longer times some population builds up in the other Zeeman sublevel that is not directly coupled to g〉 by the laser (blue dashed line), resulting in a slowly increasing excess of Rydberg fraction similar to what is observed experimentally for some configurations.
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Labuhn, H., Barredo, D., Ravets, S. et al. Tunable twodimensional arrays of single Rydberg atoms for realizing quantum Ising models. Nature 534, 667–670 (2016). https://doi.org/10.1038/nature18274
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