Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Exploiting disorder to probe spin and energy hydrodynamics

Nature Physics volume 19, pages 1027–1032 (2023)Cite this article

Subjects

Abstract

An outstanding challenge in large-scale quantum platforms is to simultaneously achieve strong interactions, giving rise to the most interesting behaviours, and local addressing, which can probe them. In the context of correlated phases, local addressing allows one to directly probe the nature of the system’s order. At the same time, such addressing allows the study of quantum information spreading and operator growth in out-of-equilibrium scenarios. Here we introduce a technique that enables the measurement of local correlation functions, down to single-site resolution, despite access to only global controls. Our approach leverages the intrinsic disorder present in a solid-state spin ensemble to dephase the non-local components of the correlation function. Utilizing this toolset, we measure both the spin and energy transport in nuclear spin chains. By tuning the interaction Hamiltonian via Floquet engineering, we investigate the cross-over between ballistic and diffusive hydrodynamics. Interestingly, in certain parameter regimes, we observe the coexistence of diffusive spin transport with ballistic energy transport, a hallmark of interacting integrable systems.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Measuring local autocorrelations by utilizing global control and intrinsic on-site disorder.
Fig. 2: Experimental verification of random initial state preparation.
Fig. 3: Observing different universality classes of hydrodynamics.
Fig. 4: Finite-time effect of transport in the presence of an on-site random field.

Similar content being viewed by others

Data availability

Source data are provided with this paper. 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. Halliwell, J. Decoherent histories and the emergent classicality of local densities. Phys. Rev. Lett. 83, 2481–2485 (1999).

    MathSciNet  MATH  ADS  Google Scholar 

  2. Wyatt, R. E. Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics Vol. 28 (Springer Science & Business Media, 2005).

  3. Hartle, J. B. The quasiclassical realms of this quantum universe. Found. Phys. 41, 982–1006 (2011).

    MathSciNet  MATH  ADS  Google Scholar 

  4. Spohn, H. Large Scale Dynamics of Interacting Particles (Springer Science & Business Media, 2012).

  5. Birkhoff, G. in Hydrodynamics (Princeton University Press, 2015).

  6. De Nardis, J., Bernard, D. & Doyon, B. Hydrodynamic diffusion in integrable systems. Phys. Rev. Lett. 121, 160603 (2018).

    MathSciNet  Google Scholar 

  7. Andreev, A., Kivelson, S. A. & Spivak, B. Hydrodynamic description of transport in strongly correlated electron systems. Phys. Rev. Lett. 106, 256804 (2011).

    ADS  Google Scholar 

  8. Žnidarič, M., Scardicchio, A. & Varma, V. K. Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system. Phys. Rev. Lett. 117, 040601 (2016).

    ADS  Google Scholar 

  9. Bertini, B., Collura, M., De Nardis, J. & Fagotti, M. Transport in out-of-equilibrium x x z chains: exact profiles of charges and currents. Phys. Rev. Lett. 117, 207201 (2016).

    ADS  Google Scholar 

  10. Leviatan, E., Pollmann, F., Bardarson, J. H., Huse, D. A. & Altman, E. Quantum thermalization dynamics with matrix-product states. Preprint at arXiv:1702.08894 (2017).

  11. Ye, B., Machado, F., White, C. D., Mong, R. S. & Yao, N. Y. Emergent hydrodynamics in nonequilibrium quantum systems. Phys. Rev. Lett. 125, 030601 (2020).

    ADS  Google Scholar 

  12. Ljubotina, M., Žnidarič, M. & Prosen, T. Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet. Phys. Rev. Lett. 122, 210602 (2019).

    ADS  Google Scholar 

  13. Ye, B., Machado, F., Kemp, J., Hutson, R. B. & Yao, N. Y. Universal Kardar-Parisi-Zhang dynamics in integrable quantum systems. Phys. Rev. Lett. 129, 230602 (2022).

    MathSciNet  ADS  Google Scholar 

  14. Sommer, A., Ku, M., Roati, G. & Zwierlein, M. W. Universal spin transport in a strongly interacting Fermi gas. Nature 472, 201–204 (2011).

    ADS  Google Scholar 

  15. Moll, P. J., Kushwaha, P., Nandi, N., Schmidt, B. & Mackenzie, A. P. Evidence for hydrodynamic electron flow in PdCoO2. Science 351, 1061–1064 (2016).

    ADS  Google Scholar 

  16. Cepellotti, A. et al. Phonon hydrodynamics in two-dimensional materials. Nat. Commun. 6, 6400 (2015).

    ADS  Google Scholar 

  17. Crossno, J. et al. Observation of the dirac fluid and the breakdown of the wiedemann-franz law in graphene. Science 351, 1058–1061 (2016).

    ADS  Google Scholar 

  18. Agarwal, K., Gopalakrishnan, S., Knap, M., Müller, M. & Demler, E. Anomalous diffusion and griffiths effects near the many-body localization transition. Phys. Rev. Lett. 114, 160401 (2015).

    ADS  Google Scholar 

  19. Castro-Alvaredo, O. A., Doyon, B. & Yoshimura, T. Emergent hydrodynamics in integrable quantum systems out of equilibrium. Phys. Rev. X 6, 041065 (2016).

    Google Scholar 

  20. Bertini, B. Finite-temperature transport in one-dimensional quantum lattice models. Rev. Mod. Phys. 93, 025003 (2021).

    MathSciNet  ADS  Google Scholar 

  21. Ilievski, E. & De Nardis, J. Microscopic origin of ideal conductivity in integrable quantum models. Phys. Rev. Lett. 119, 020602 (2017).

    ADS  Google Scholar 

  22. Gopalakrishnan, S. & Vasseur, R. Kinetic theory of spin diffusion and superdiffusion in x x z spin chains. Phys. Rev. Lett. 122, 127202 (2019).

    ADS  Google Scholar 

  23. De Nardis, J., Bernard, D. & Doyon, B. Diffusion in generalized hydrodynamics and quasiparticle scattering. SciPost Phys 6, 049 (2019).

    MathSciNet  ADS  Google Scholar 

  24. Ilievski, E., De Nardis, J., Gopalakrishnan, S., Vasseur, R. & Ware, B. Superuniversality of superdiffusion. Phys. Rev. X 11, 031023 (2021).

    Google Scholar 

  25. De Nardis, J., Gopalakrishnan, S., Vasseur, R. & Ware, B. Stability of superdiffusion in nearly integrable spin chains. Phys. Rev. Lett. 127, 057201 (2021).

    MathSciNet  ADS  Google Scholar 

  26. Friedman, A. J., Gopalakrishnan, S. & Vasseur, R. Diffusive hydrodynamics from integrability breaking. Phys. Rev. B 101, 180302 (2020).

    ADS  Google Scholar 

  27. Schemmer, M., Bouchoule, I., Doyon, B. & Dubail, J. Generalized hydrodynamics on an atom chip. Phys. Rev. Lett. 122, 090601 (2019).

    ADS  Google Scholar 

  28. Zu, C. et al. Emergent hydrodynamics in a strongly interacting dipolar spin ensemble. Nature 597, 45–50 (2021).

    ADS  Google Scholar 

  29. Malvania, N. et al. Generalized hydrodynamics in strongly interacting 1d bose gases. Science 373, 1129–1133 (2021).

    MathSciNet  MATH  ADS  Google Scholar 

  30. Wei, D. et al. Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion. Science 376, 716–720 (2022).

    ADS  Google Scholar 

  31. Joshi, M. K. et al. Observing emergent hydrodynamics in a long-range quantum magnet. Science 376, 720–724 (2022).

    ADS  Google Scholar 

  32. Martin, L. S. et al. Controlling local thermalization dynamics in a Floquet-engineered dipolar ensemble. Preprint at arXiv:2209.09297 (2022).

  33. Altman, E. et al. Quantum simulators: architectures and opportunities. PRX Quantum 2, 017003 (2021).

    Google Scholar 

  34. Bakr, W. S., Gillen, J. I., Peng, A., Folling, S. & Greiner, M. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 74–77 (2009).

    ADS  Google Scholar 

  35. Zhang, W. & Cory, D. First direct measurement of the spin diffusion rate in a homogenous solid. Phys. Rev. Lett. 80, 1324 (1998).

    ADS  Google Scholar 

  36. Rittweger, E., Han, K. Y., Irvine, S. E., Eggeling, C. & Hell, S. W. STED microscopy reveals crystal colour centres with nanometric resolution. Nat. Photonics 3, 144–147 (2009).

    ADS  Google Scholar 

  37. Maurer, P. et al. Far-field optical imaging and manipulation of individual spins with nanoscale resolution. Nat. Phys. 6, 912–918 (2010).

    Google Scholar 

  38. Chen, E. H., Gaathon, O., Trusheim, M. E. & Englund, D. Wide-field multispectral super-resolution imaging using spin-dependent fluorescence in nanodiamonds. Nano Lett. 13, 2073–2077 (2013).

    ADS  Google Scholar 

  39. Pfender, M., Aslam, N., Waldherr, G., Neumann, P. & Wrachtrup, J. Single-spin stochastic optical reconstruction microscopy. Proc. Natl Acad. Sci. USA 111, 14669–14674 (2014).

    ADS  Google Scholar 

  40. Arai, K. et al. Fourier magnetic imaging with nanoscale resolution and compressed sensing speed-up using electronic spins in diamond. Nat. Nanotechnol. 10, 859–864 (2015).

    ADS  Google Scholar 

  41. Hunt, G. A. Some theorems concerning brownian motion. Trans. Am. Math. Soc. 81, 294–319 (1956).

    MathSciNet  MATH  Google Scholar 

  42. Waugh, J. S., Huber, L. M. & Haeberlen, U. Approach to high-resolution nmr in solids. Phys. Rev. Lett. 20, 180–182 (1968).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  44. Haeberlen, U. & Waugh, J. S. Coherent averaging effects in magnetic resonance. Phys. Rev. 175, 453–467 (1968).

    ADS  Google Scholar 

  45. Peng, P. et al. Deep reinforcement learning for quantum hamiltonian engineering. Phys. Rev. Appl. 18, 024033 (2022).

    ADS  Google Scholar 

  46. Grabowski, M. & Mathieu, P. Structure of the conservation laws in quantum integrable spin chains with short range interactions. Ann. Phys. 243, 299–371 (1995).

    MathSciNet  MATH  ADS  Google Scholar 

  47. Zotos, X., Naef, F. & Prelovsek, P. Transport and conservation laws. Phys. Rev. B 55, 11029–11032 (1997).

    ADS  Google Scholar 

  48. Klümper, A. & Johnston, D. Thermodynamics of the spin-1/2 antiferromagnetic uniform Heisenberg chain. Phys. Rev. Lett. 84, 4701–4704 (2000).

    ADS  Google Scholar 

  49. Sakai, K. & Klümper, A. Non-dissipative thermal transport in the massive regimes of the xxz chain. J. Phys. A: Math. Gen. 36, 11617–11629 (2003).

    MathSciNet  MATH  ADS  Google Scholar 

  50. Prosen, T. & Žnidarič, M. Matrix product simulations of non-equilibrium steady states of quantum spin chains. J. Stat. Mech.: Theory Exp. 2009, P02035 (2009).

    Google Scholar 

  51. Steinigeweg, R. & Gemmer, J. Density dynamics in translationally invariant spin-1 2 chains at high temperatures: a current-autocorrelation approach to finite time and length scales. Phys. Rev. B 80, 184402 (2009).

    ADS  Google Scholar 

  52. Žnidarič, M. Spin transport in a one-dimensional anisotropic Heisenberg model. Phys. Rev. Lett. 106, 220601 (2011).

    ADS  Google Scholar 

  53. Karrasch, C., Moore, J. & Heidrich-Meisner, F. Real-time and real-space spin and energy dynamics in one-dimensional spin-1 2 systems induced by local quantum quenches at finite temperatures. Phys. Rev. B 89, 075139 (2014).

    ADS  Google Scholar 

  54. Lucioni, E. et al. Observation of subdiffusion in a disordered interacting system. Phys. Rev. Lett. 106, 230403 (2011).

    ADS  Google Scholar 

  55. Vosk, R., Huse, D. A. & Altman, E. Theory of the many-body localization transition in one-dimensional systems. Phys. Rev. X 5, 031032 (2015).

    Google Scholar 

  56. Potter, A. C., Vasseur, R. & Parameswaran, S. Universal properties of many-body delocalization transitions. Phys. Rev. X 5, 031033 (2015).

    Google Scholar 

  57. Sahay, R., Machado, F., Ye, B., Laumann, C. R. & Yao, N. Y. Emergent ergodicity at the transition between many-body localized phases. Phys. Rev. Lett. 126, 100604 (2021).

    ADS  Google Scholar 

  58. Zaburdaev, V., Denisov, S. & Klafter, J. Lévy walks. Rev. Mod. Phys. 87, 483–530 (2015).

    ADS  Google Scholar 

  59. Nahum, A., Vijay, S. & Haah, J. Operator spreading in random unitary circuits. Phys. Rev. X 8, 021014 (2018).

    Google Scholar 

  60. Von Keyserlingk, C., Rakovszky, T., Pollmann, F. & Sondhi, S. L. Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws. Phys. Rev. X 8, 021013 (2018).

    Google Scholar 

  61. Rakovszky, T. & Pollmann, F. & Von Keyserlingk, C. Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation. Phys. Rev. X 8, 031058 (2018).

    Google Scholar 

  62. Khemani, V., Vishwanath, A. & Huse, D. A. Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws. Phys. Rev. X 8, 031057 (2018).

    Google Scholar 

  63. Xu, S. & Swingle, B. Accessing scrambling using matrix product operators. Nat. Phys. 16, 199–204 (2020).

    Google Scholar 

  64. Xu, S. & Swingle, B. Locality, quantum fluctuations, and scrambling. Phys. Rev. X 9, 031048 (2019).

    Google Scholar 

  65. Sahu, S., Xu, S. & Swingle, B. Scrambling dynamics across a thermalization-localization quantum phase transition. Phys. Rev. Lett. 123, 165902 (2019).

    ADS  Google Scholar 

  66. Schuster, T. et al. Many-body quantum teleportation via operator spreading in the traversable wormhole protocol. Phys. Rev. X 12, 031013 (2022).

    Google Scholar 

  67. Li, J. et al. Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator. Phys. Rev. X 7, 031011 (2017).

    ADS  Google Scholar 

  68. Landsman, K. A. et al. Verified quantum information scrambling. Nature 567, 61–65 (2019).

    ADS  Google Scholar 

  69. Blok, M. S. et al. Quantum information scrambling on a superconducting qutrit processor. Phys. Rev. X 11, 021010 (2021).

    Google Scholar 

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

    ADS  Google Scholar 

  71. Cappellaro, P., Ramanathan, C. & Cory, D. G. Simulations of information transport in spin chains. Phys. Rev. Lett. 99, 250506 (2007).

    ADS  Google Scholar 

  72. Cappellaro, P., Viola, L. & Ramanathan, C. Coherent-state transfer via highly mixed quantum spin chains. Phys. Rev. A 83, 032304 (2011).

    ADS  Google Scholar 

  73. 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).

    ADS  Google Scholar 

  74. Rufeil-Fiori, E., Sánchez, C. M., Oliva, F. Y., Pastawski, H. M. & Levstein, P. R. Effective one-body dynamics in multiple-quantum nmr experiments. Phys. Rev. A 79, 032324 (2009).

    ADS  Google Scholar 

  75. Zhang, W. et al. NMR multiple quantum coherences in quasi-one-dimensional spin systems: comparison with ideal spin-chain dynamics. Phys. Rev. A 80, 052323 (2009).

    ADS  Google Scholar 

  76. Comodi, P., Liu, Y., Zanazzi, P. & Montagnoli, M. Structural and vibrational behaviour of fluorapatite with pressure. Part I: in situ single-crystal x-ray diffraction investigation. Phys. Chem. Miner. 28, 219–224 (2001).

    ADS  Google Scholar 

Download references

Acknowledgements

We thank C. Ramanathan, H. Zhou, M. Leigh, N. Leitao, F. Machado, J. Kemp, J. Moore and M. Lukin for helpful conversations. This work was supported in part by the National Science Foundation under grant No. PHY1915218. P.P. thanks MathWorks for their support in the form of a Graduate Student Fellowship. The opinions and views expressed in this publication are from the authors and not necessarily from MathWorks. B.Y. acknowledges support from the Army Research Office through the MURI program (W911NF-20-1-0136). N.Y.Y. acknowledges support from the NSF through the QLCI program (OMA-2016245) and the David and Lucile Packard foundation.

Author information

Author notes

  1. These authors contributed equally: Pai Peng, Bingtian Ye.

Authors and Affiliations

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

    Pai Peng

  2. Department of Physics, University of California, Berkeley, CA, USA

    Bingtian Ye & Norman Y. Yao

  3. Department of Physics, Harvard University, Cambridge, MA, USA

    Bingtian Ye & Norman Y. Yao

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

    Paola Cappellaro

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

    Paola Cappellaro

Authors
  1. Pai Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bingtian Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Norman Y. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paola Cappellaro
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

P.P. designed and performed the experiment with assistance from P.C. B.Y. and N.Y.Y. performed the numerical and analytical calculations. P.C. supervised the project. All authors worked on the interpretation of the data and contributed to writing the manuscript.

Corresponding authors

Correspondence to Pai Peng or Paola Cappellaro.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Jianming Cai and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Ab initio calculation of disordered field and decoherence profile.

Disordered on-site field generated by 31P. a. Numerical calculation of distribution of the on-site field strength. The four-Gaussian fit gives a standard deviation of 2.217(2) krad/s for each Gaussian peak. The single-Gaussian fit gives a standard deviation of 6.05(6) krad/s. b. Left axis: Decoherence profile generated by the calculated distribution of on-site field and the single-peak Gaussian approximation. Right axis: Statistical correlation between the random amplitudes of local observables on two closest 19F. As the coherence approaches zero, the statistical correlation also vanishes.

Source data

Extended Data Fig. 2 Raw data for transport with disorder.

Spin (a) and energy (b) autocorrelation for various disorder field strength h. Data are presented as mean values +/- SD from readout noise (for additional details see the Supplementary Information).

Source data

Supplementary information

Supplementary Information

Supplementary Figs 1–4, Discussion and Table 1.

Source data

Source Data Fig. 2

Statistical source data.

Source Data Fig. 3

Statistical source data.

Source Data Fig. 4

Statistical source data.

Source Data Extended Data Fig. 1

Statistical source data.

Source Data Extended Data Fig. 2

Statistical source data.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peng, P., Ye, B., Yao, N.Y. et al. Exploiting disorder to probe spin and energy hydrodynamics. Nat. Phys. 19, 1027–1032 (2023). https://doi.org/10.1038/s41567-023-02024-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-023-02024-4

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing