Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet

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Controllable arrays of ions and ultracold atoms can simulate complex many-body phenomena and may provide insights into unsolved problems in modern science. To this end, experimentally feasible protocols for quantifying the buildup of quantum correlations and coherence are needed, as performing full state tomography does not scale favourably with the number of particles. Here we develop and experimentally demonstrate such a protocol, which uses time reversal of the many-body dynamics to measure out-of-time-order correlation functions (OTOCs) in a long-range Ising spin quantum simulator with more than 100 ions in a Penning trap. By measuring a family of OTOCs as a function of a tunable parameter we obtain fine-grained information about the state of the system encoded in the multiple quantum coherence spectrum, extract the quantum state purity, and demonstrate the buildup of up to 8-body correlations. Future applications of this protocol could enable studies of many-body localization, quantum phase transitions, and tests of the holographic duality between quantum and gravitational systems.

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Figure 1: Illustration of the many-body echo scheme.
Figure 2: Phonon-mediated, reversible spin–spin coupling in a Penning trap.
Figure 3: Measured fidelity and coherence spectrum of N = 48 ions.
Figure 4: Probing scrambling through magnetization dynamics.


  1. 1

    Loschmidt, J. Über den Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft. 1. Teil. Sitzungsber. Kais. Akad. Wiss. Wien, Math. Naturwiss. Classe 73, 128–142 (1876).

  2. 2

    Hahn, E. L. Spin echoes. Phys. Rev. 80, 580–594 (1950).

  3. 3

    Baum, J., Munowitz, M., Garroway, A. N. & Pines, A. Multiple-quantum dynamics in solid-state NMR. J. Chem. Phys. 83, 2015–2025 (1985).

  4. 4

    Widera, A. et al. Quantum spin dynamics of mode-squeezed Luttinger liquids in two-component atomic gases. Phys. Rev. Lett. 100, 140401 (2008).

  5. 5

    Linnemann, D. et al. Quantum-enhanced sensing based on time reversal of non-linear dynamics. Phys. Rev. Lett. 117, 013001 (2016).

  6. 6

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

  7. 7

    Yao, N. Y. et al. Interferometric approach to probing fast scrambling. Preprint at (2016).

  8. 8

    Shen, H., Zhang, P., Fan, R. & Zhai, H. Out-of-time-order correlation at a quantum phase transition. Preprint at (2016).

  9. 9

    Zhu, G., Hafezi, M. & Grover, T. Measurement of many-body chaos using a quantum clock. Phys. Rev. A 94, 062329 (2016).

  10. 10

    Sekino, Y. & Susskind, L. Fast scramblers. J. High Energy Phys. 2008, 065 (2008).

  11. 11

    Shenker, S. H. & Stanford, D. Black holes and the butterfly effect. J. High Energy Phys. 2014, 067 (2014).

  12. 12

    Shenker, S. H. & Stanford, D. Stringy effects in scrambling. J. High Energy Phys. 2015, 132 (2015).

  13. 13

    Eisert, J., Friesdorf, M. & Gogolin, C. Quantum many-body systems out of equilibrium. Nat. Phys. 11, 124–130 (2015).

  14. 14

    Fu, W. & Sachdev, S. Numerical study of fermion and boson models with infinite-range random interactions. Phys. Rev. B 94, 035135 (2016).

  15. 15

    Danshita, I., Hanada, M. & Tezuka, M. Creating and probing the Sachdev-Ye-Kitaev model with ultracold gases: towards experimental studies of quantum gravity. Preprint at (2016).

  16. 16

    Hosur, P., Qi, X.-L., Roberts, D. A. & Yoshida, B. Chaos in quantum channels. J. High Energy Phys. 2016, 004 (2016).

  17. 17

    Kitaev, A. A simple model of quantum holography (7 April 2015 and 27 May 2015). Talks given at The Kavli Institute for Theoretical Physics (KITP) (University of California).

  18. 18

    Bohnet, J. G. et al. Quantum spin dynamics and entanglement generation with hundreds of trapped ions. Science 352, 1297–1301 (2016).

  19. 19

    Leibfried, D. et al. Creation of a six-atom ‘Schrödinger cat’ state. Nature 438, 639–642 (2005).

  20. 20

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

  21. 21

    Strobel, H. et al. Fisher information and entanglement of non-gaussian spin states. Science 345, 424–427 (2014).

  22. 22

    Álvarez, G. A., Suter, D. & Kaiser, R. Localization-delocalization transition in the dynamics of dipolar-coupled nuclear spins. Science 349, 846–848 (2015).

  23. 23

    Sánchez, C. M., Levstein, P. R., Acosta, R. H. & Chattah, A. K. NMR Loschmidt echoes as quantifiers of decoherence in interacting spin systems. Phys. Rev. A 80, 012328 (2009).

  24. 24

    Debnath, S. et al. Demonstration of a small programmable quantum computer with atomic qubits. Nature 536, 63–66 (2016).

  25. 25

    Cucchietti, F. M. Time reversal in an optical lattice. J. Opt. Soc. Am. B 27, A30–A35 (2010).

  26. 26

    Leroux, I. D., Schleier-Smith, M. H. & Vuletić, V. Implementation of cavity squeezing of a collective atomic spin. Phys. Rev. Lett. 104, 073602 (2010).

  27. 27

    Hosten, O., Krishnakumar, R., Engelsen, N. J. & Kasevich, M. A. Quantum phase magnification. Science 352, 1552–1555 (2016).

  28. 28

    Douglas, J. S. et al. Quantum many-body models with cold atoms coupled to photonic crystals. Nat. Photon. 9, 326–331 (2015).

  29. 29

    Macrí, T., Pezzé, L. & Smerzi, A. Loschmidt echo for quantum metrology. Phys. Rev. A 94, 010102(R) (2016).

  30. 30

    Houck, A. A., Tureci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nat. Phys. 8, 292–299 (2012).

  31. 31

    Sørensen, A. & Mølmer, K. Quantum computation with ions in thermal motion. Phys. Rev. Lett. 82, 1971–1974 (1999).

  32. 32

    Britton, J. W. et al. Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins. Nature 484, 489–492 (2012).

  33. 33

    Sawyer, B. C. et al. Spectroscopy and thermometry of drumhead modes in a mesoscopic trapped-ion crystal using entanglement. Phys. Rev. Lett. 108, 213003 (2012).

  34. 34

    Biercuk, M. J. et al. High-fidelity quantum control using ion crystals in a Penning trap. Quant. Info. Comput. 9, 920–949 (2009).

  35. 35

    Leibfried, D. et al. Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate. Nature 422, 412–415 (2003).

  36. 36

    Britton, J. W. et al. Vibration-induced field fluctuations in a superconducting magnet. Phys. Rev. A 93, 062511 (2016).

  37. 37

    Davis, E., Bentsen, G. & Schleier-Smith, M. Approaching the Heisenberg limit without single-particle detection. Phys. Rev. Lett. 116, 053601 (2016).

  38. 38

    Cucchietti, F. M., Fernandez-Vidal, S. & Paz, J. P. Universal decoherence induced by an environmental quantum phase transition. Phys. Rev. A 75, 032337 (2007).

  39. 39

    Quan, H. T., Song, Z., Liu, X. F., Zanardi, P. & Sun, C. P. Decay of Loschmidt echo enhanced by quantum criticality. Phys. Rev. Lett. 96, 140604 (2006).

  40. 40

    Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).

  41. 41

    Jacquod, P. & Petitjean, C. Decoherence, entanglement and irreversibility in quantum dynamical systems with few degrees of freedom. Adv. Phys. 58, 67–196 (2009).

  42. 42

    Fröwis, F., Sekatski, P. & Dür, W. Detecting large quantum Fisher information with finite measurement precision. Phys. Rev. Lett. 116, 090801 (2016).

  43. 43

    Li, J. et al. Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator scrambling. Preprint at (2016).

  44. 44

    Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004).

  45. 45

    Richerme, P. et al. Non-local propagation of correlations in quantum systems with long-range interactions. Nature 511, 198–201 (2014).

  46. 46

    Jurcevic, P. et al. Quasiparticle engineering and entanglement propagation in a quantum many-body system. Nature 511, 202–205 (2014).

  47. 47

    Uys, H. et al. Decoherence due to elastic Rayleigh scattering. Phys. Rev. Lett. 105, 200401 (2010).

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We thank P. Hauke, J. Price and S. Kolkowitz for discussions and careful reading of our manuscript, and gratefully acknowledge J. Britton and B. Sawyer for preceding experimental contributions to this work. Supported by Defense Advanced Research Projects Agency (DARPA) ATN program through grant number W911NF-16-1-0576 through the Army Research Office (ARO), NSF grant PHY 1521080, JILA-NSF grant PFC-1125844, the ARO, and the Air Force Office of Scientific Research and its Multidisciplinary University Research Initiative (AFOSR-MURI) (A.M.R.) and by a National Research Council Research Associateship Award at NIST (J.G.B. and M.L.W.). All authors acknowledge financial support from NIST.

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J.G.B. and J.J.B. conducted the experiment. The theoretical modelling was done by M.G., A.S.-N., M.L.W. and A.M.R. All authors jointly interpreted and discussed the experimental data.

Correspondence to Ana Maria Rey.

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The authors declare no competing financial interests.

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Gärttner, M., Bohnet, J., Safavi-Naini, A. et al. Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet. Nature Phys 13, 781–786 (2017) doi:10.1038/nphys4119

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