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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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References

  1. Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012)

    Article  CAS  Google Scholar 

  2. Ladd, T. D. et al. Quantum computers. Nature 464, 45–53 (2010)

    Article  ADS  CAS  Google Scholar 

  3. Monroe, C. & Kim, J. Scaling the ion trap quantum processor. Science 339, 1164–1169 (2013)

    Article  ADS  CAS  Google Scholar 

  4. Devoret, M. H. & Schoelkopf, R. J. Superconducting circuits for quantum information: an outlook. Science 339, 1169–1174 (2013)

    Article  ADS  CAS  Google Scholar 

  5. Awschalom, D. D., Bassett, L. C., Dzurak, A. S., Hu, E. L. & Petta, J. R. Quantum spintronics: engineering and manipulating atom-like spins in semiconductors. Science 339, 1174–1179 (2013)

    Article  ADS  CAS  Google Scholar 

  6. Monz, T. 14-qubit entanglement: creation and coherence. Phys. Rev. Lett. 106, 130506 (2011)

    Article  ADS  Google Scholar 

  7. Islam, R. et al. Emergence and frustration of magnetism with variable-range interactions in a quantum simulator. Science 340, 583–587 (2013)

    Article  ADS  CAS  Google Scholar 

  8. Song, C. et al. 10-qubit entanglement and parallel logic operations with a superconducting circuit. Phys. Rev. Lett. 119, 180511 (2017)

    Article  ADS  Google Scholar 

  9. Gärttner, M. et al. Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet. Nat. Phys. 13, 781–786 (2017)

    Article  Google Scholar 

  10. Kuhr, S. Quantum-gas microscopes: a new tool for cold-atom quantum simulators. Natl Sci. Rev. 3, 170–172 (2016)

    Article  CAS  Google Scholar 

  11. Trotzky, S. et al. Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas. Nat. Phys. 8, 325–330 (2012)

    Article  CAS  Google Scholar 

  12. Mazurenko, A. et al. A cold-atom Fermi-Hubbard antiferromagnet. Nature 545, 462–466 (2017)

    Article  ADS  CAS  Google Scholar 

  13. Rønnow, T. et al. Defining and detecting quantum speedup. Science 345, 420–424 (2014)

    Article  ADS  Google Scholar 

  14. McMahon, P. L. et al. A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 614–617 (2016)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  15. Jaksch, D. et al. Fast quantum gates for neutral atoms. Phys. Rev. Lett. 85, 2208–2211 (2000)

    Article  ADS  CAS  Google Scholar 

  16. Weimer, H., Müller, M., Lesanovsky, I., Zoller, P. & Büchler, H. P. A Rydberg quantum simulator. Nat. Phys. 6, 382–388 (2010)

    Article  CAS  Google Scholar 

  17. Wilk, T. et al. Entanglement of two individual neutral atoms using Rydberg blockade. Phys. Rev. Lett. 104, 010502 (2010)

    Article  ADS  CAS  Google Scholar 

  18. Isenhower, L. et al. Demonstration of a neutral atom controlled-NOT quantum gate. Phys. Rev. Lett. 104, 010503 (2010)

    Article  ADS  CAS  Google Scholar 

  19. Saffman, M. Quantum computing with atomic qubits and Rydberg interactions: progress and challenges. J. Phys. B 49, 202001 (2016)

    Article  ADS  Google Scholar 

  20. Pritchard, J. D. et al. Cooperative atom-light interaction in a blockaded Rydberg ensemble. Phys. Rev. Lett. 105, 193603 (2010)

    Article  ADS  CAS  Google Scholar 

  21. Schauß, P. et al. Observation of spatially ordered structures in a two-dimensional Rydberg gas. Nature 491, 87–91 (2012)

    Article  ADS  Google Scholar 

  22. Schauß, P. et al. Crystallization in Ising quantum magnets. Science 347, 1455–1458 (2015)

    Article  ADS  Google Scholar 

  23. Zeiher, J. et al. Coherent many-body spin dynamics in a long-range interacting Ising chain. Preprint at https://arxiv.org/abs/1705.08372 (2017)

  24. Labuhn, H. et al. Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models. Nature 534, 667–670 (2016)

    Article  ADS  CAS  Google Scholar 

  25. Barredo, D., de Léséleuc, S., Lienhard, V., Lahaye, T. & Browaeys, A. An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays. Science 354, 1021–1023 (2016)

    Article  ADS  CAS  Google Scholar 

  26. Endres, M. et al. Atom-by-atom assembly of defect-free one-dimensional cold atom arrays. Science 354, 1024–1027 (2016)

    Article  ADS  CAS  Google Scholar 

  27. Kim, H. et al. In situ single-atom array synthesis using dynamic holographic optical tweezers. Nat. Commun. 7, 13317 (2016)

    Article  ADS  CAS  Google Scholar 

  28. Dudin, Y. O., Li, L., Bariani, F. & Kuzmich, A. Observation of coherent many-body Rabi oscillations. Nat. Phys. 8, 790–794 (2012)

    Article  CAS  Google Scholar 

  29. Zeiher, J. et al. Microscopic characterization of scalable coherent Rydberg superatoms. Phys. Rev. X 5, 031015 (2015)

    Google Scholar 

  30. Fendley, P., Sengupta, K. & Sachdev, S. Competing density-wave orders in a one-dimensional hard-boson model. Phys. Rev. B 69, 075106 (2004)

    Article  ADS  Google Scholar 

  31. Pohl, T., Demler, E. & Lukin, M. D. Dynamical crystallization in the dipole blockade of ultracold atoms. Phys. Rev. Lett. 104, 043002 (2010)

    Article  ADS  CAS  Google Scholar 

  32. Petrosyan, D., Mølmer, K. & Fleischhauer, M. On the adiabatic preparation of spatially-ordered Rydberg excitations of atoms in a one-dimensional optical lattice by laser frequency sweeps. J. Phys. B 49, 084003 (2016)

    Article  ADS  Google Scholar 

  33. Schachenmayer, J., Lesanovsky, I., Micheli, A. & Daley, A. J. Dynamical crystal creation with polar molecules or Rydberg atoms in optical lattices. New J. Phys. 12, 103044 (2010)

    Article  ADS  Google Scholar 

  34. Richerme, P. et al. Experimental performance of a quantum simulator: Optimizing adiabatic evolution and identifying many-body ground states. Phys. Rev. A 88, 012334 (2013)

    Article  ADS  Google Scholar 

  35. Sachdev, S. Quantum Phase Transitions 2nd edn (Cambridge Univ. Press, 2009)

  36. Sachdev, S., Sengupta, K. & Girvin, S. M. Mott insulators in strong electric fields. Phys. Rev. B 66, 075128 (2002)

    Article  ADS  Google Scholar 

  37. Zurek, W. H., Dorner, U. & Zoller, P. Dynamics of a quantum phase transition. Phys. Rev. Lett. 95, 105701 (2005)

    Article  ADS  Google Scholar 

  38. D’Alessio, L., Kafri, Y., Polkovnikov, A. & Rigol, M. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Adv. Phys. 65, 239–362 (2016)

    Article  ADS  Google Scholar 

  39. Abanin, D., De Roeck, W., Ho, W. W. & Huveneers, F. A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems. Commun. Math. Phys. 354, 809–827 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  40. Feiguin, A. et al. Interacting anyons in topological quantum liquids: the golden chain. Phys. Rev. Lett. 98, 160409 (2007)

    Article  ADS  Google Scholar 

  41. Lesanovsky, I. & Katsura, H. Interacting Fibonacci anyons in a Rydberg gas. Phys. Rev. A 86, 041601 (2012)

    Article  ADS  Google Scholar 

  42. Moessner, R. & Raman, K. S. in Introduction to Frustrated Magnetism (eds Lacroix, C. et al.) 437–479 (Springer, 2011)

  43. Lester, B. J., Luick, N., Kaufman, A. M., Reynolds, C. M. & Regal, C. A. Rapid production of uniformly filled arrays of neutral atoms. Phys. Rev. Lett. 115, 073003 (2015)

    Article  ADS  Google Scholar 

  44. Pichler, H., Zhu, G., Seif, A., Zoller, P. & Hafezi, M. Measurement protocol for the entanglement spectrum of cold atoms. Phys. Rev. X 6, 041033 (2016)

    Google Scholar 

  45. Schiulaz, M., Silva, A. & Müller, M. Dynamics in many-body localized quantum systems without disorder. Phys. Rev. B 91, 184202 (2015)

    Article  ADS  Google Scholar 

  46. Swingle, B., Bentsen, G., Schleier-Smith, M. & Hayden, P. Measuring the scrambling of quantum information. Phys. Rev. A 94, 040302(R) (2016)

    Article  ADS  MathSciNet  Google Scholar 

  47. Chandran, A., Schulz, M. D. & Burnell, F. J. The eigenstate thermalization hypothesis in constrained Hilbert spaces: a case study in non-abelian anyon chains. Phys. Rev. B 94, 235122 (2016)

    Article  ADS  Google Scholar 

  48. Huse, D. A. & Fisher, M. E. Commensurate melting, domain walls, and dislocations. Phys. Rev. B 29, 239–270 (1984)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  49. Lechner, W., Hauke, P. & Zoller, P. A quantum annealing architecture with all-to-all connectivity from local interactions. Sci. Adv. 1, e1500838 (2015)

  50. Farhi, E. & Harrow, A. W. Quantum supremacy through the quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1602.07674 (2016)

  51. Singer, K., Stanojevic, J., Weidemüller, M. & Côté, R. Long-range interactions between alkali Rydberg atom pairs correlated to the ns–ns, np–np and nd–nd asymptotes. J. Phys. B 38, S295 (2005)

    Article  ADS  CAS  Google Scholar 

  52. Hall, J. L. & Zhu, M. in Laser Manipulation of Atoms and Ions (eds Arimondo, E. et al.) 671–702 (Elsevier, 1993)

  53. Fox, R. W., Oates, C. W. & Hollberg, L. W. in Cavity-Enhanced Spectroscopies (eds van Zee, R. D. & Looney, J. P. ) 1–46 (Elsevier, 2003)

  54. Beterov, I. I., Ryabtsev, I. I., Tretyakov, D. B. & Entin, V. M. Quasiclassical calculations of blackbody-radiation-induced depopulation rates and effective lifetimes of Rydberg nS, nP, and nD alkali-metal atoms with n ≤ 80. Phys. Rev. A 79, 052504 (2009)

    Article  ADS  Google Scholar 

  55. Glaz, J. & Sison, C. P. Simultaneous confidence intervals for multinomial proportions. J. Stat. Plan. Inference 82, 251–262 (1999)

    Article  MathSciNet  Google Scholar 

  56. Johansson, J. R., Nation, P. D. & Nori, F. QuTiP: an open-source Python framework for the dynamics of open quantum systems. Comput. Phys. Commun. 183, 1760–1772 (2012)

    Article  ADS  CAS  Google Scholar 

  57. Boixo, S. et al. Characterizing quantum supremacy in near-term devices. Preprint at https://arxiv.org/abs/1608.00263 (2016)

  58. Baxter, R. J. Exactly Solved Models in Statistical Mechanics (Courier Corporation, 2007)

  59. Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  60. Haegeman, J. Time-dependent variational principle for quantum lattices. Phys. Rev. Lett. 107, 070601 (2011)

    Article  ADS  Google Scholar 

  61. Vidal, G. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett. 93, 040502 (2004)

    Article  ADS  Google Scholar 

  62. Daley, A. J., Kollath, C., Schollwöck, U. & Vidal, G. Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces. J. Stat. Mech. 2004, P04005 (2004)

    Article  Google Scholar 

Download references

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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