Spin models are the prime example of simplified many-body Hamiltonians used to model complex, strongly correlated real-world materials1. 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 simulation2 of spin Hamiltonians using the tools of atomic and molecular physics has become a very active field over the past years, using ultracold atoms3 or molecules4 in optical lattices, or trapped ions5. 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 two-dimensional arrays of optical microtraps with arbitrary geometries, where filling fractions range from 60 to 100 per cent. When excited to high-energy Rydberg D states, the atoms undergo strong interactions whose anisotropic character opens the way to simulating exotic matter6. We illustrate the versatility of our system by studying the dynamics of a quantum Ising-like spin-1/2 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 spin-1/2 system, while for strongly anisotropic situations the multilevel structure of the D states has a measurable influence7,8. Our findings establish arrays of single Rydberg atoms as a versatile platform for the study of quantum magnetism.
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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 (FET-Open Xtrack Project HAIRS, H2020 FET-PROACT Project RySQ, and EU Marie-Curie Program ITN COHERENCE FP7-PEOPLE-2010-ITN-265031 (H.L.)), by the ‘PALM’ Labex (project QUANTICA) and by the Région Île-de-France in the framework of DIM Nano-K.
The authors declare no competing financial interests.
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
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 TN when the triggering criterion is set to N = Nt: achieving full loading requires an exponentially long time, limiting the method in practice to Nt ≤ 9. The triggering times can vary substantially depending on the density of the magneto-optical 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 pN of having a number N of loaded traps in the partially loaded regime for the 30-trap ‘racetrack’ (left) and the 49-trap 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.
a, Experimental determination of ε. From the data of the full blockade experiments (Fig. 2 of main text), we plot the probability P0 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).
a, Fully loaded arrays of 1 to 9 traps (n = 82). b, Partially loaded array of Nt = 19 traps, containing from N = 10 to N = 15 atoms (n = 100). The column on the left shows the probability P0 of recapturing all atoms, the centre column the probability P1 of losing just one atom out of N, and the column on the right the probability P2 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 P0 data as a function of the number of atoms in the array. Error bars, s.d.
a, For Ωτ = 3.1, we observe strongly contrasted oscillations in the pair correlation function g2(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 30-trap, racetrack-shaped 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 Rb = 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, fR approaches, at long times, the close-packing limit a/Rb of hard rods of length Rb (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) 30-trap array. b, Pair correlation function g(2)(k) for Ωτ ≈ 2.0, for the partially loaded (solid line) or fully loaded (thin dashed line) 30-trap 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 eight-atom ring.
Calculated pair correlation function g(2)(k) as a function of the excitation time for the eight-atom 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 re-establishing 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 fR.
We use the toy model of nP1/2 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 spin-1/2 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 two-dimensional 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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