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Floquet prethermalization in dipolar spin chains

Nature Physics volume 17, pages 444–447 (2021)Cite this article

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Abstract

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.

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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.

References

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

    Article  Google Scholar 

  2. Lazarides, A., Das, A. & Moessner, R. Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90, 012110 (2014).

    Article  ADS  Google Scholar 

  3. D’Alessio, L. & Rigol, M. Long-time behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X 4, 041048 (2014).

    Google Scholar 

  4. Kim, H., Ikeda, T. N. & Huse, D. A. Testing whether all eigenstates obey the eigenstate thermalization hypothesis. Phys. Rev. E 90, 052105 (2014).

    Article  ADS  Google Scholar 

  5. Abanin, D. A., De Roeck, W., Ho, W. W. & Huveneers, F. Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems. Phys. Rev. B 95, 014112 (2017).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  7. Kuwahara, T., Mori, T. & Saito, K. Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems. Ann. Phys. 367, 96–124 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Abanin, D., De Roeck, W., Ho, W. W. & Huveneers, F. A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems. Commun. Math. Phys. 354, 809–827 (2017).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Gómez-León, A. & Platero, G. Floquet–Bloch theory and topology in periodically driven lattices. Phys. Rev. Lett. 110, 200403 (2013).

    Article  ADS  Google Scholar 

  10. Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011).

    Article  Google Scholar 

  11. Wang, Y., Steinberg, H., Jarillo-Herrero, P. & Gedik, N. Observation of Floquet–Bloch states on the surface of a topological insulator. Science 342, 453–457 (2013).

    Article  ADS  Google Scholar 

  12. Eckardt, A., Weiss, C. & Holthaus, M. Superfluid–insulator transition in a periodically driven optical lattice. Phys. Rev. Lett. 95, 260404 (2005).

    Article  ADS  Google Scholar 

  13. Tsuji, N., Oka, T., Werner, P. & Aoki, H. Dynamical band flipping in fermionic lattice systems: an ac-field-driven change of the interaction from repulsive to attractive. Phys. Rev. Lett. 106, 236401 (2011).

    Article  ADS  Google Scholar 

  14. Görg, F. Enhancement and sign change of magnetic correlations in a driven quantum many-body system. Nature 553, 481–485 (2018).

    Article  ADS  Google Scholar 

  15. Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Childs, A. M., Maslov, D., Nam, Y., Ross, N. J. & Su, Y. Toward the first quantum simulation with quantum speedup. Proc. Natl Acad. Sci. USA 115, 9456–9461 (2018).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  19. Bastidas, V. M., Emary, C., Regler, B. & Brandes, T. Nonequilibrium quantum phase transitions in the Dicke model. Phys. Rev. Lett. 108, 043003 (2012).

    Article  ADS  Google Scholar 

  20. Wei, K. X., Ramanathan, C. & Cappellaro, P. Exploring localization in nuclear spin chains. Phys. Rev. Lett. 120, 070501 (2018).

    Article  ADS  Google Scholar 

  21. Lazarides, A., Das, A. & Moessner, R. Fate of many-body localization under periodic driving. Phys. Rev. Lett. 115, 030402 (2015).

    Article  ADS  Google Scholar 

  22. Ponte, P., Papić, Z., Huveneers, F. & Abanin, D. A. Many-body localization in periodically driven systems. Phys. Rev. Lett. 114, 140401 (2015).

    Article  ADS  Google Scholar 

  23. Bordia, P., Lüschen, H., Schneider, U., Knap, M. & Bloch, I. Periodically driving a many-body localized quantum system. Nat. Phys. 13, 460–464 (2017).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  25. Heyl, M., Hauke, P. & Zoller, P. Quantum localization bounds Trotter errors in digital quantum simulation. Sci. Adv. 5, eaau8342 (2019).

    Article  ADS  Google Scholar 

  26. Sieberer, L. M. Digital quantum simulation, Trotter errors, and quantum chaos of the kicked top. npj Quantum Inf. 5, 78 (2019).

    Article  ADS  Google Scholar 

  27. Ji, K. & Fine, B. V. Suppression of heating in quantum spin clusters under periodic driving as a dynamic localization effect. Phys. Rev. Lett. 121, 050602 (2018).

    Article  ADS  Google Scholar 

  28. D’Alessio, L. & Polkovnikov, A. Many-body energy localization transition in periodically driven systems. Ann. Phys. 333, 19–33 (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Yin, C., Peng, P., Huang, X., Ramanathan, C. & Cappellaro, P. Prethermal quasiconserved observables in Floquet quantum systems. Preprint at https://arxiv.org/abs/2005.11150 (2020).

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

    Google Scholar 

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

    Google Scholar 

  32. 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).

    Google Scholar 

  33. Herrmann, A., Murakami, Y., Eckstein, M. & Werner, P. Floquet prethermalization in the resonantly driven Hubbard model. Europhys. Lett. 120, 57001 (2018).

    Article  ADS  Google Scholar 

  34. Bukov, M., Heyl, M., Huse, D. A. & Polkovnikov, A. Heating and many-body resonances in a periodically driven two-band system. Phys. Rev. B 93, 155132 (2016).

    Article  ADS  Google Scholar 

  35. Mallayya, K. & Rigol, M. Heating rates in periodically driven strongly interacting quantum many-body systems. Phys. Rev. Lett. 123, 240603 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  36. Goldman, M. Spin Temperature and NMR in Solids (Clarendon, 1970).

  37. Wei, K. X. Emergent prethermalization signatures in out-of-time ordered correlations. Phys. Rev. Lett. 123, 090605 (2019).

    Article  ADS  Google Scholar 

  38. Jeener, J. & Broekaert, P. Nuclear magnetic resonance in solids: thermodynamic effects of a pair of rf pulses. Phys. Rev. 157, 232–240 (1967).

    Article  ADS  Google Scholar 

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

    Google Scholar 

  40. Ramanathan, C., Cappellaro, P., Viola, L. & Cory, D. G. Experimental characterization of coherent magnetization transport in a one-dimensional spin system. New J. Phys. 13, 103015 (2011).

    Article  ADS  Google Scholar 

  41. Magnus, W. On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  42. Blanes, S., Casas, F., Oteo, J. & Ros, J. The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  43. Bukov, M., D’Alessio, L. & Polkovnikov, A. Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Adv. Phys. 64, 139–226 (2015).

    Article  ADS  Google Scholar 

  44. Yan, F. Rotating-frame relaxation as a noise spectrum analyser of a superconducting qubit undergoing driven evolution. Nat. Commun. 4, 2337 (2013).

    Article  ADS  Google Scholar 

  45. Laucht, A. A dressed spin qubit in silicon. Nat. Nanotechnol. 12, 61–66 (2017).

    Article  ADS  Google Scholar 

  46. Golter, D. A., Baldwin, T. K. & Wang, H. Protecting a solid-state spin from decoherence using dressed spin states. Phys. Rev. Lett. 113, 237601 (2014).

    Article  ADS  Google Scholar 

  47. Timoney, N. Quantum gates and memory using microwave-dressed states. Nature 476, 185–188 (2011).

    Article  ADS  Google Scholar 

  48. Munowitz, M. & Pines, A. in Advances in Chemical Physics Vol. 66 (eds Prigogine, I. & Rice, S.) 1–152 (Wiley, 1975).

  49. Gärttner, M., Hauke, P. & Rey, A. M. Relating out-of-time-order correlations to entanglement via multiple-quantum coherences. Phys. Rev. Lett. 120, 040402 (2018).

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

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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Author notes

  1. These authors contributed equally: Pai Peng, Chao Yin.

Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, USA

    Pai Peng

  2. Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Chao Yin, Xiaoyang Huang & Paola Cappellaro

  3. Department of Physics and Astronomy, Dartmouth College, Hanover, NH, USA

    Chandrasekhar Ramanathan

  4. Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

    Paola Cappellaro

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  1. Pai Peng
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Contributions

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.

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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). https://doi.org/10.1038/s41567-020-01120-z

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