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# Critical prethermal discrete time crystal created by two-frequency driving

## Abstract

Discrete time crystals are non-equilibrium many-body phases of matter characterized by spontaneously broken discrete time-translation symmetry under periodic driving. At sufficiently high driving frequencies, the system enters the Floquet prethermalization regime, in which the periodically driven many-body state has a lifetime vastly exceeding the intrinsic decay time of the system. Here, we report the observation of long-lived prethermal discrete time-crystalline order in a three-dimensional (3D) lattice of 13C nuclei in diamond at room temperature. We demonstrate a two-frequency driving protocol, involving an interleaved application of slow and fast drives that simultaneously prethermalize the spins with an emergent quasi-conserved magnetization along the $${\hat{{{{\bf{x}}}}}}$$ axis. This enables continuous and highly resolved observation of their dynamic evolution. We obtain videos of the time-crystalline response with a clarity and throughput orders of magnitude greater than previous experiments. Parametric control over the drive frequencies allows us to reach time-crystal lifetimes of up to 396 Floquet cycles, which we measure in a single-shot experiment. Such rapid measurement enables detailed characterization of the entire phase diagram, highlighting the role of prethermalization in stabilizing the time-crystal response. The two-frequency drive approach expands the toolkit for investigating non-equilibrium phases of matter stabilized by emergent quasi-conservation laws.

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## Data availability

Data from experiments and simulations displayed in the main text are available in Zenodo with the identifier https://doi.org/10.5281/zenodo.7301638. All other data from the Supplementary Information are available from the authors upon reasonable request.

## References

1. Sacha, K. & Zakrzewski, J. Time crystals: a review. Rep. Prog. Phys. 81, 016401 (2017).

2. Khemani, V., Moessner, R. & Sondhi, S. A brief history of time crystals. Preprint at https://arxiv.org/abs/1910.10745 (2019).

3. Else, D. V., Monroe, C., Nayak, C. & Yao, N. Y. Discrete time crystals. Annu. Rev. Condens. Matter Phys. 11, 467–499 (2020).

4. Sacha, K. Modeling spontaneous breaking of time-translation symmetry. Phys. Rev. A 91, 033617 (2015).

5. Khemani, V., Lazarides, A., Moessner, R. & Sondhi, S. L. Phase structure of driven quantum systems. Phys. Rev. Lett. 116, 250401 (2016).

6. Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016).

7. von Keyserlingk, C. W., Khemani, V. & Sondhi, S. L. Absolute stability and spatiotemporal long-range order in floquet systems. Phys. Rev. B 94, 085112 (2016).

8. Yao, N. Y., Potter, A. C., Potirniche, I.-D. & Vishwanath, A. Discrete time crystals: rigidity, criticality and realizations. Phys. Rev. Lett. 118, 030401 (2017).

9. Ho, W. W., Choi, S., Lukin, M. D. & Abanin, D. A. Critical time crystals in dipolar systems. Phys. Rev. Lett. 119, 010602 (2017).

10. Liao, L., Smits, J., van der Straten, P. & Stoof, H. T. C. Dynamics of a space-time crystal in an atomic Bose-Einstein condensate. Phys. Rev. A 99, 013625 (2019).

11. Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system. Nature 543, 221–225 (2017).

12. Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).

13. Pal, S., Nishad, N., Mahesh, T. S. & Sreejith, G. J. Temporal order in periodically driven spins in star-shaped clusters. Phys. Rev. Lett. 120, 180602 (2018).

14. Smits, J., Liao, L., Stoof, H. T. C. & van der Straten, P. Observation of a space-time crystal in a superfluid quantum gas. Phys. Rev. Lett. 121, 185301 (2018).

15. Rovny, J., Blum, R. L. & Barrett, S. E. 31P NMR study of discrete time-crystalline signatures in an ordered crystal of ammonium dihydrogen phosphate. Phys. Rev. B 97, 184301 (2018).

16. Randall, J. et al. Many-body-localized discrete time crystal with a programmable spin-based quantum simulator. Science 374, 1474–1478 (2021).

17. Mi, X. et al. Time-crystalline eigenstate order on a quantum processor. Nature 601, 531–536 (2021).

18. Singh, K. et al. Quantifying and controlling prethermal nonergodicity in interacting Floquet matter. Phys. Rev. X 9, 041021 (2019).

19. Rubio-Abadal, A. et al. Floquet prethermalization in a Bose-Hubbard system. Phys. Rev. X 10, 021044 (2020).

20. Peng, P., Yin, C., Huang, X., Ramanathan, C. & Cappellaro, P. Floquet prethermalization in dipolar spin chains. Nat. Phys. 17, 444–447 (2021).

21. Abanin, D. A., De Roeck, W. & Huveneers, F. Exponentially slow heating in periodically driven many-body systems. Phys. Rev. Lett. 115, 256803 (2015).

22. Mori, T., Kuwahara, T. & Saito, K. Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems. Phys. Rev. Lett. 116, 120401 (2016).

23. Machado, F., Else, D. V., Kahanamoku-Meyer, G. D., Nayak, C. & Yao, N. Y. Long-range prethermal phases of nonequilibrium matter. Phys. Rev. X 10, 011043 (2020).

24. Else, D. V., Bauer, B. & Nayak, C. Prethermal phases of matter protected by time-translation symmetry. Phys. Rev. X 7, 011026 (2017).

25. Pizzi, A., Nunnenkamp, A. & Knolle, J. Classical prethermal phases of matter. Phys. Rev. Lett. 127, 140602 (2021).

26. Ye, B., Machado, F. & Yao, N. Y. Floquet phases of matter via classical prethermalization. Phys. Rev. Lett. 127, 140603 (2021).

27. Yao, N. Y., Nayak, C., Balents, L. & Zaletel, M. P. Classical discrete time crystals. Nat. Phys. 16, 438–447 (2020).

28. Rovny, J., Blum, R. L. & Barrett, S. E. Observation of discrete-time-crystal signatures in an ordered dipolar many-body system. Phys. Rev. Lett. 120, 180603 (2018).

29. Kyprianidis, A. et al. Observation of a prethermal discrete time crystal. Science 372, 1192–1196 (2021).

30. Ajoy, A. et al. Orientation-independent room temperature optical 13C hyperpolarization in powdered diamond. Sci. Adv. 4, eaar5492 (2018).

31. Ajoy, A. et al. Enhanced dynamic nuclear polarization via swept microwave frequency combs. Proc. Natl Acad. Sci. USA 115, 10576–10581 (2018).

32. Duer, M. Introduction to Solid-State NMR Spectroscopy (Wiley, 2004).

33. Beatrez, W. et al. Floquet prethermalization with lifetime exceeding 90 s in a bulk hyperpolarized solid. Phys. Rev. Lett. 127, 170603 (2021).

34. Reynhardt, E. Spin lattice relaxation of spin-1/2 nuclei in solids containing diluted paramagnetic impurity centers. I. Zeeman polarization of nuclear spin system. Concepts Magn. Reson. A 19A, 20–35 (2003).

35. Ajoy, A. et al. Hyperpolarized relaxometry based nuclear T1 noise spectroscopy in diamond. Nat. Commun. 10, 5160 (2019).

36. Luitz, D. J., Moessner, R., Sondhi, S. L. & Khemani, V. Prethermalization without temperature. Phys. Rev. X 10, 021046 (2020).

37. Video of full dataset from Fig. 2a of main text https://youtu.be/61ZqLgbCuyo (2021).

38. Video of full dataset from Fig. 2b and Fig. 3a of main text (first 55 Floquet cycles) https://youtu.be/m5iASnBZ9oo (2021).

39. Weinberg, P. & Bukov, M. QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems part I: spin chains. SciPost Phys. 2, 003 (2017).

## Acknowledgements

We thank J. Bardarson, M. Heyl, C. von Keyserlingk, D. Luitz, R. Moessner, J. Reimer and D. Suter for valuable discussions. A. Ajoy acknowledges funding from ONR under contract no. N00014-20-1-2806. C.F. acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement nos. 679722 and 101001902). M.B. was supported by the Marie Skłodowska-Curie grant agreement no. 890711, and the Bulgarian National Science Fund within National Science Program VIHREN, contract no. KP-06-DV-5 (until 25 June 2021). The computational work reported on in this Article was enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC), partially funded by the Swedish Research Council through grant agreement no. 2018-05973 and the Würzburg HPC cluster. Computational work reported on in this Article was performed on the Würzburg HPC cluster.

## Author information

Authors

### Contributions

W.B., C.F., M.B. and A. Ajoy conceived the research. W.B., A.P., E.d.L.S., A. Akkiraju, J.D.A., S.C., P.R., E.D. and A. Ajoy set up the experimental apparatus, performed measurements and analysed the data. C.F. performed the numerical simulations and the perturbative analysis. A. Ajoy and M.B. supervised the experiment and the theory work.

### Corresponding authors

Correspondence to Marin Bukov or Ashok Ajoy.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Physics thanks Tim Hugo Taminiau, Fedor Jelezko and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary Information

Supplementary text, Table 1 and Figs. 1–19.

### Supplementary Video 1

Video of zoomed in view of three Floquet cycles, showing prethermalizing dynamics during two-frequency driving.

### Supplementary Video 2

Video of 25 Floquet cycles, showing emergence of the prethermal DTC.

### Supplementary Video 3

Video of 55 Floquet cycles, showing emergence of the prethermal DTC.

### Supplementary Video 4

Video of 155 Floquet cycles, showing emergence of the prethermal DTC.

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Reprints and Permissions

Beatrez, W., Fleckenstein, C., Pillai, A. et al. Critical prethermal discrete time crystal created by two-frequency driving. Nat. Phys. 19, 407–413 (2023). https://doi.org/10.1038/s41567-022-01891-7

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• DOI: https://doi.org/10.1038/s41567-022-01891-7