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Critical prethermal discrete time crystal created by two-frequency driving


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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Fig. 1: Experimental Implementation.
Fig. 2: Continuously observed prethermalization and PDTC.
Fig. 3: Prethermal DTC phase diagram.
Fig. 4: Experimental characterization of PDTC rigidity and prethermal lifetimes.
Fig. 5: Experimental characterization of the heating dynamics at small N.

Data availability

Data from experiments and simulations displayed in the main text are available in Zenodo with the identifier All other data from the Supplementary Information are available from the authors upon reasonable request.


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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.

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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.

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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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Beatrez, W., Fleckenstein, C., Pillai, A. et al. Critical prethermal discrete time crystal created by two-frequency driving. Nat. Phys. 19, 407–413 (2023).

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