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Probing many-body dynamics on a 51-atom quantum simulator

Abstract

Controllable, coherent many-body systems can provide insights into the fundamental properties of quantum matter, enable the realization of new quantum phases and could ultimately lead to computational systems that outperform existing computers based on classical approaches. Here we demonstrate a method for creating controlled many-body quantum matter that combines deterministically prepared, reconfigurable arrays of individually trapped cold atoms with strong, coherent interactions enabled by excitation to Rydberg states. We realize a programmable Ising-type quantum spin model with tunable interactions and system sizes of up to 51 qubits. Within this model, we observe phase transitions into spatially ordered states that break various discrete symmetries, verify the high-fidelity preparation of these states and investigate the dynamics across the phase transition in large arrays of atoms. In particular, we observe robust many-body dynamics corresponding to persistent oscillations of the order after a rapid quantum quench that results from a sudden transition across the phase boundary. Our method provides a way of exploring many-body phenomena on a programmable quantum simulator and could enable realizations of new quantum algorithms.

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Figure 1: Experimental platform.
Figure 2: Phase diagram and build-up of crystalline phases.
Figure 3: Comparison with a fully coherent simulation.
Figure 4: Scaling behaviour.
Figure 5: Quantifying Z2 order in a 51-atom array after a slow detuning sweep.
Figure 6: Emergent oscillations in many-body dynamics after sudden quench.

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Acknowledgements

We thank E. Demler, A. Chandran, S. Sachdev, A. Vishwanath, P. Zoller, P. Silvi, T. Pohl, M. Knap, M. Fleischhauer, S. Hofferberth and A. Harrow for discussions. This work was supported by NSF, CUA, ARO, and a Vannevar Bush Faculty Fellowship. H.B. acknowledges support by a Rubicon Grant of the Netherlands Organization for Scientific Research (NWO). A.O. acknowledges support by a research fellowship from the German Research Foundation (DFG). S.S. acknowledges funding from the European Union under the Marie Skłodowska Curie Individual Fellowship Programme H2020-MSCA-IF-2014 (project number 658253). H.P. acknowledges support by the National Science Foundation (NSF) through a grant at the Institute for Theoretical Atomic Molecular and Optical Physics (ITAMP) at Harvard University and the Smithsonian Astrophysical Observatory. H.L. acknowledges support by the National Defense Science and Engineering Graduate (NDSEG) Fellowship.

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Contributions

The experiments and data analysis were carried out by H.B., S.S., A.K., H.L., A.O., A.S.Z. and M.E. Theoretical analysis was performed by H.P. and S.C. All work was supervised by M.G., V.V. and M.D.L. All authors discussed the results and contributed to the manuscript.

Corresponding authors

Correspondence to Markus Greiner, Vladan Vuletić or Mikhail D. Lukin.

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Extended data figures and tables

Extended Data Figure 1 Experimental sequence and Rydberg laser set-up.

a, The tweezer array is initially loaded from a magneto-optical trap (MOT). A single-site-resolved fluorescence image taken with an electron-multiplying CCD camera (EMCCD) is used to identify the loaded traps. Using this information, a feedback protocol rearranges the loaded atoms into a preprogrammed configuration, which is verified by the second EMCCD image. After that, all atoms are optically pumped into the |F = 2, mF = −2〉 state, the tweezers are turned off and the Rydberg lasers are pulsed. After the traps are turned back on, a third EMCCD image is taken to detect Rydberg excitations with single-site resolution. b, Schematic representation of the Rydberg laser set-up, which is used to stabilize two external cavity diode lasers to a reference optical cavity with a fast Pound–Drever–Hall lock. TA, tapered amplifier; AOM, acousto-optic modulator; EOM, electo-optic modulator; PD, photodetector; PBS, polarizing beam splitter; QWP, quarter-wave plate.

Extended Data Figure 2 Drop-recapture curve.

Measurements of atom loss probability as a function of trap-off time. For short times of up to 4 μs, the loss is dominated by finite trap lifetime (1% plateau). At larger trap-off times, the atomic motion away from the tweezer introduces additional losses. The solid line is a Monte Carlo simulation for a temperature of 11.8 μK.

Extended Data Figure 3 Typical Rabi oscillation, homogeneity and coherence for non-interacting atoms.

a = 23 μm, . a, Rabi oscillations. We observe a typical decay time of about 6 μs, which is limited mainly by intensity fluctuations from shot to shot. b, The fitted Rabi frequency for each atom across the array (spatial extent of about 300 μm) is homogeneous to within 3%. c, Measurement of the population in the Rydberg state after a spin echo pulse sequence (inset). We find no decay of coherence over typical measurement periods of several microseconds, thereby ruling out fast sources of decoherence. Error bars in ac denote 68% confidence intervals.

Extended Data Figure 4 Spectroscopic measurement of Rydberg interactions.

Spectroscopy on pairs of atoms separated by approximately 5.74 μm is shown. a, For single-atom losses, we observe a single peak at Δ = 0 corresponding to the two-photon coupling from |g, g〉 to |W〉. b, For two-atom losses, we observe an additional peak at Δ = 2π × 12.2 MHz. This corresponds to the four-photon coupling from |g, g〉 to |r, r〉 through the intermediate state |W〉, detuned by Δ. The interaction energy is then V = 2Δ. This four-photon resonance is broadened as a result of random atom positions within the optical tweezers that result in fluctuations in interaction strengths from shot to shot of the experiment. Solid lines are fits with a single Lorentzian (a) and the sum of two Lorentzians (b). Error bars denote 68% confidence intervals.

Extended Data Figure 5 Ground-state preparation probability.

We compare the ground-state preparation probability obtained here (measured, red circles; corrected for detection infidelity, blue circles) with the most complete previous observations of a Z2-symmetry breaking transition in a system of trapped ions (green circles)34. We note that the interaction Hamiltonians for the two systems are not identical, owing to the finite interaction range. In particular, the long-range interactions tend to frustrate adiabatic transitions into Z2-ordered states in ref. 34 and, to lesser extent, in this work. Error bars denote 68% confidence intervals.

Extended Data Figure 6 State preparation with 51-atom clusters.

a, Average position-dependent Rydberg probability in a 51-atom cluster after the adiabatic sweep. The Z2 order is visible at the edges of the system, whereas the presence of domain walls leads to an apparently featureless bulk throughout the centre of the system. Inset, average Rydberg probabilities in a 13-atom chain, in which the Z2 order is visible throughout the system, but the small system size prevents the study of bulk properties. b, Variance of the domain-wall distribution during Z2 state preparation. Points and error bars represent measured values. The solid red line corresponds to a full numerical simulation of the dynamics using a matrix product state ansatz (see text and Fig. 5). Error bars in a and b denote 68% confidence intervals.

Extended Data Figure 7 State reconstruction.

a, Reconstructed parent distribution. b, Comparison of measured domain-wall distribution (red) and predicted observation given the parent distribution in a (blue). c, Difference between the two distributions in b.

Extended Data Figure 8 Comparison to a thermal state.

a, Domain-wall density for thermal states at different entropy per atom s/kB. The lower line corresponds to the actual number of domain walls in a system of the corresponding temperature; the upper line gives the domain-wall density that would be measured at this temperature, given the finite detection fidelity. The horizontal dashed line denotes the experimentally measured domain-wall density, from which we infer a corresponding entropy per atom, and equivalently temperature, in a thermal ensemble. b, Entropy per atoms for a thermal state at given inverse temperature β = 1/(kBT) in a 51-atom array. c, Expected distribution of the number of domain walls for the thermal ensemble at β = 3.44/Δ, with (red) and without (blue) taking into account finite detection fidelity. d, Experimentally measured correlation function g(2)(d) (blue) and correlation function corresponding to a thermal ensemble at β = 3.44/Δ (grey). The inset shows the rectified correlation function on a logarithmic scale, indicating that the measured correlation function decays exponentially, but with a different correlation length from that obtained from a thermal state with the measured number of domain walls.

Extended Data Figure 9 Oscillations in domain-wall density using a variational matrix product state ansatz.

The dynamics of the domain-wall density in the bulk of the array under the constrained Hamiltonian c at Δ = 0 is shown. The blue line shows the evolution of the domain-wall density obtained by integrating the variational equations of motion (equation (5)) with initial conditions θa = π/2, θb = 0, that is, the crystalline initial state. The red line shows the exact dynamics of the domain-wall density at the centre of a system of 25 atoms initially in the crystalline state under the constrained Hamiltonian c.

Extended Data Figure 10 Decay of oscillations after a quench and entropy growth.

a, Dynamics of the domain-wall density under the constrained Hamiltonian c for different initial states. The red line shows the domain-wall density for a system of 25 atoms initially prepared in the electronic ground state. In this case, the domain-wall density relaxes quickly to a steady value corresponding to thermalization. In contrast, the blue line shows the dynamics if the system is initialized in the Z2-ordered state. In this case, the domain-wall density oscillates over several periods and even for very long times does not relax fully to a steady value. b, Same as in a, but taking into account the full 1/R6 interactions. While the dynamics for an initial state |gN is very similar to the one obtained in the constrained case, for the crystalline initial state the decay of the oscillations is faster than in the constrained model. c, Growth of entanglement entropy in a bipartite splitting of the 25-atom array for the different cases displayed in a and b. The entropy is defined as the von Neumann entropy of the reduced state of the first 13 atoms of the array. The dashed lines correspond to dynamics under the constrained Hamiltonian, neglecting the 1/R6 tail, whereas the solid lines take the full interactions into account. Red lines correspond to the initial state |gN, whereas blue lines correspond to crystalline initial states. In all panels we chose Ω = 2π × 2 MHz and, where applicable, interaction parameters such that the nearest-neighbour interaction evaluates to Vi,i+1 = 2π × 25.6 MHz.

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Bernien, H., Schwartz, S., Keesling, A. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017). https://doi.org/10.1038/nature24622

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