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

## Abstract

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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## Acknowledgements

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.

## Author information

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.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

## Supplementary information

### Supplementary information

Supplementary information (PDF 628 kb)

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Reprints and Permissions

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

• #### DOI

https://doi.org/10.1038/nphys4119

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