Abstract
Controllable, coherent manybody 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 manybody 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 Isingtype 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 highfidelity preparation of these states and investigate the dynamics across the phase transition in large arrays of atoms. In particular, we observe robust manybody 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 manybody phenomena on a programmable quantum simulator and could enable realizations of new quantum algorithms.
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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 H2020MSCAIF2014 (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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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.
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Extended data figures and tables
Extended Data Figure 1 Experimental sequence and Rydberg laser setup.
a, The tweezer array is initially loaded from a magnetooptical trap (MOT). A singlesiteresolved fluorescence image taken with an electronmultiplying 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, m_{F} = −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 singlesite resolution. b, Schematic representation of the Rydberg laser setup, 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, acoustooptic modulator; EOM, electooptic modulator; PD, photodetector; PBS, polarizing beam splitter; QWP, quarterwave plate.
Extended Data Figure 2 Droprecapture curve.
Measurements of atom loss probability as a function of trapoff time. For short times of up to 4 μs, the loss is dominated by finite trap lifetime (1% plateau). At larger trapoff 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 noninteracting 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 a–c 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 singleatom losses, we observe a single peak at Δ = 0 corresponding to the twophoton coupling from g, g〉 to W〉. b, For twoatom losses, we observe an additional peak at Δ = 2π × 12.2 MHz. This corresponds to the fourphoton coupling from g, g〉 to r, r〉 through the intermediate state W〉, detuned by Δ. The interaction energy is then V = 2Δ. This fourphoton 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 Groundstate preparation probability.
We compare the groundstate preparation probability obtained here (measured, red circles; corrected for detection infidelity, blue circles) with the most complete previous observations of a Z_{2}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 longrange interactions tend to frustrate adiabatic transitions into Z_{2}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 51atom clusters.
a, Average positiondependent Rydberg probability in a 51atom cluster after the adiabatic sweep. The Z_{2} 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 13atom chain, in which the Z_{2} order is visible throughout the system, but the small system size prevents the study of bulk properties. b, Variance of the domainwall distribution during Z_{2} 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 domainwall 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, Domainwall density for thermal states at different entropy per atom s/k_{B}. The lower line corresponds to the actual number of domain walls in a system of the corresponding temperature; the upper line gives the domainwall density that would be measured at this temperature, given the finite detection fidelity. The horizontal dashed line denotes the experimentally measured domainwall 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/(k_{B}T) in a 51atom 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 domainwall density using a variational matrix product state ansatz.
The dynamics of the domainwall density in the bulk of the array under the constrained Hamiltonian ℋ_{c} at Δ = 0 is shown. The blue line shows the evolution of the domainwall density obtained by integrating the variational equations of motion (equation (5)) with initial conditions θ_{a} = π/2, θ_{b} = 0, that is, the crystalline initial state. The red line shows the exact dynamics of the domainwall density at the centre of a system of 25 atoms initially in the crystalline state under the constrained Hamiltonian ℋ_{c}.
Extended Data Figure 10 Decay of oscillations after a quench and entropy growth.
a, Dynamics of the domainwall density under the constrained Hamiltonian ℋ_{c} for different initial states. The red line shows the domainwall density for a system of 25 atoms initially prepared in the electronic ground state. In this case, the domainwall 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 Z_{2}ordered state. In this case, the domainwall 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/R^{6} interactions. While the dynamics for an initial state g〉^{⊗N} 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 25atom 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/R^{6} tail, whereas the solid lines take the full interactions into account. Red lines correspond to the initial state g〉^{⊗N}, 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 nearestneighbour interaction evaluates to V_{i,i+1} = 2π × 25.6 MHz.
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Bernien, H., Schwartz, S., Keesling, A. et al. Probing manybody dynamics on a 51atom quantum simulator. Nature 551, 579–584 (2017). https://doi.org/10.1038/nature24622
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DOI: https://doi.org/10.1038/nature24622
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