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Floquet prethermalization in dipolar spin chains


Periodically driven Floquet quantum systems could provide a promising platform to investigate novel physics out of equilibrium1, but the drive generically heats the system to a featureless infinite-temperature state2,3,4. Fortunately, for high driving frequency, the heat absorption rate has been theoretically predicted to be exponentially small, giving rise to a long-lived prethermal regime that exhibits all the intriguing properties of Floquet systems5,6,7,8. Here we experimentally observe Floquet prethermalization using NMR techniques and probe the heating rate. We first show the relaxation of a far-from-equilibrium initial state to a long-lived prethermal state, well described by a time-independent ‘prethermal’ Hamiltonian. By measuring the autocorrelation of this prethermal Hamiltonian we can further experimentally confirm the predicted exponentially slow heating rate. More strikingly, we find that, on the timescale at which the prethermal Hamiltonian picture breaks down, the Floquet system still possesses other quasiconservation laws. Our results demonstrate that it is possible to realize robust Floquet engineering, thus enabling the experimental observation of non-trivial Floquet phases of matter.

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Fig. 1: Schematic of the experimental system and Floquet prethermalization.
Fig. 2: Breakdown of energy conservation and Floquet heating rate.
Fig. 3: Robustness of dipolar order.

Data availability

Source data and all other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


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We thank H. Zhou, W.-J. Zhang and Z. Li for discussion. This work was supported in part by the National Science Foundation under grants no. PHY1734011, no. PHY1915218 and no. OIA-1921199.

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Authors and Affiliations



P.P. performed the experiments. P.P. and C.Y. analysed the data and developed the theoretical modelling. P.C. supervised the project. All authors discussed the results and contributed to the manuscript.

Corresponding authors

Correspondence to Pai Peng or Paola Cappellaro.

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Extended data

Extended Data Fig. 1 Background decay dependence on Jτ.

Decay rate of 〈Y(n)Y〉 (blue) and 〈Dy(n)Dy〉 (green) under engineered dipolar Hamiltonian JDy as a function of Jτ. The range of Jτ studied was obtained by varying the scaling u (Supplementary Information) from 0.098 to 0.646, while keeping fixed τ = 120μs. In the inset, we compare the background decay rates with the Floquet decay rates (dashed lines).

Extended Data Fig. 2 Signatures of a robust quasiconserved quantity for the kicked dipolar model.

We numerically evaluate the expansion of the quasiconserved quantity Dpre and plot the norm of each term (normalized by L2L) as a function of the expansion order m, for various hτ = Jτ. Jτ varies from 0 (light colors) to 2 (dark colors) in steps of 0.2. (a) Infidelity 1 − 〈Dpre()Dpre〉/〈DpreDpre〉 of the infinite-time averaged Dpre as a function of the order, m, for various hτ = Jτ. Jτ varies from 0 (light colors) to 2 (dark colors) in steps of 0.2. (b) Infidelity 1 − 〈Hpre()Hpre〉/〈HpreHpre〉 of the infinite-time averaged prethermal Hamiltonian Hpre as a function of the order, m, for various hτ = Jτ. L = 12 was used in (a-b). The normalized autocorrelation of Hpre converges to 1 in a smaller parameter range (Jτ 1) than Dpre (Jτ 1.6) (c) Fidelities of the two conserved quantities (〈Hpre()Hpre〉/〈HpreHpre〉 and 〈Dpre()Dpre〉/〈DpreDpre〉) evaluated to 7th order as a function of hτ for three different system sizes. The fidelities show a notable drop when L is increased from 8 to 12 at Jτ 1.8 for Dpre and Jτ 1.2 for Hpre, again indicating that Dpre is more robust than Hpre.

Supplementary information

Supplementary Information

Supplementary Figs. 1–4 and Discussion.

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Peng, P., Yin, C., Huang, X. et al. Floquet prethermalization in dipolar spin chains. Nat. Phys. 17, 444–447 (2021).

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