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.

  • Letter
  • Published:

Thermalization and its mechanism for generic isolated quantum systems

An Erratum to this article was published on 11 January 2012

Abstract

An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive. Recently, meaningful experimental studies1,2 of the problem have become possible, stimulating theoretical interest3,4,5,6,7. In generic isolated systems, non-equilibrium dynamics is expected8,9 to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics. However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible10. For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete11. Some recent studies4,5 even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems. Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription. Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch12 and Srednicki13. A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages—any eigenstate in the microcanonical energy window will do, because they all give the same result.

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

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

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

Figure 1: Relaxation dynamics.
Figure 2: Thermalization in classical versus quantum mechanics.
Figure 3: Eigenstate thermalization hypothesis.
Figure 4: Temporal versus quantum fluctuations.

Similar content being viewed by others

References

  1. Kinoshita, T., Wenger, T. & Weiss, D. S. A quantum Newton's cradle. Nature 440, 900–903 (2006)

    Article  CAS  ADS  Google Scholar 

  2. Hofferberth, S., Lesanovsky, I., Fischer, B., Schumm, T. & Schmiedmayer, J. Non-equilibrium coherence dynamics in one-dimensional Bose gases. Nature 449, 324–327 (2007)

    Article  CAS  ADS  Google Scholar 

  3. Rigol, M., Dunjko, V., Yurovsky, V. & Olshanii, M. Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1d lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007)

    Article  ADS  Google Scholar 

  4. Kollath, C., Läuchli, A. & Altman, E. Quench dynamics and nonequilibrium phase diagram of the Bose-Hubbard model. Phys. Rev. Lett. 98, 180601 (2007)

    Article  ADS  Google Scholar 

  5. Manmana, S. R., Wessel, S., Noack, R. M. & Muramatsu, A. Strongly correlated fermions after a quantum quench. Phys. Rev. Lett. 98, 210405 (2007)

    Article  CAS  ADS  Google Scholar 

  6. Burkov, A. A., Lukin, M. D. & Demler, E. Decoherence dynamics in low-dimensional cold atom interferometers. Phys. Rev. Lett. 98, 200404 (2007)

    Article  CAS  ADS  Google Scholar 

  7. Calabrese, P. & Cardy, J. Quantum quenches in extended systems. J. Stat. Mech. P06008 (2007)

  8. Sengupta, K., Powell, S. & Sachdev, S. Quench dynamics across quantum critical points. Phys. Rev. A 69, 053616 (2004)

    Article  ADS  Google Scholar 

  9. Berges, J., Borsányi, S. & Wetterich, C. Prethermalization. Phys. Rev. Lett. 93, 142002 (2004)

    Article  CAS  ADS  Google Scholar 

  10. Gallavotti, G. Statistical Mechanics: A Short Treatise (Springer, Berlin, 1999)

    Book  Google Scholar 

  11. Krylov, N. S. Works on the Foundation of Statistical Physics (Princeton Univ. Press, Princeton, 1979)

    Google Scholar 

  12. Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046–2049 (1991)

    Article  CAS  ADS  Google Scholar 

  13. Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994)

    Article  CAS  ADS  Google Scholar 

  14. Sutherland, B. Beautiful Models 27–30 (World Scientific, Singapore, 2004)

    Book  Google Scholar 

  15. Rigol, M., Muramatsu, A. & Olshanii, M. Hard-core bosons on optical superlattices: Dynamics and relaxation in the superfluid and insulating regimes. Phys. Rev. A 74, 053616 (2006)

    Article  ADS  Google Scholar 

  16. Cazalilla, M. A. Effect of suddenly turning on interactions in the Luttinger model. Phys. Rev. Lett. 97, 156403 (2006)

    Article  CAS  ADS  Google Scholar 

  17. Jin, D. S., Ensher, J. R., Matthews, M. R., Wieman, C. E. & Cornell, E. A. Collective excitations of a Bose-Einstein condensate in a dilute gas. Phys. Rev. Lett. 77, 420–423 (1996)

    Article  CAS  ADS  Google Scholar 

  18. Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002)

    Article  CAS  ADS  Google Scholar 

  19. Mandel, O. et al. Controlled collisions for multi-particle entanglement of optically trapped atoms. Nature 425, 937–940 (2003)

    Article  CAS  ADS  Google Scholar 

  20. Horoi, M., Zelevinsky, V. & Brown, B. A. Chaos vs thermalization in the nuclear shell model. Phys. Rev. Lett. 74, 5194–5197 (1995)

    Article  CAS  ADS  Google Scholar 

  21. Shnirelman, A. I. Ergodic properties of eigenfunctions. Usp. Mat. Nauk 29, 181–182 (1974)

    MathSciNet  Google Scholar 

  22. Voros, A. Stochastic Behavior in Classical and Quantum Hamiltonian Systems (Springer, Berlin, 1979)

    Google Scholar 

  23. de Verdière, Y. C. Ergodicité et fonctions propres du Laplacien. Commun. Math. Phys. 102, 497–502 (1985)

    Article  ADS  Google Scholar 

  24. Zelditch, S. Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987)

    Article  MathSciNet  Google Scholar 

  25. Heller, E. J. & Landry, B. R. Statistical properties of many particle eigenfunctions. J. Phys. A 40, 9259–9274 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  26. Berry, M. V. Regular and irregular semiclassical wavefunctions. J. Phys. A 10, 2083–2091 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  27. Korepin, V. E., Bogoliubov, N. M. & Izergin, A. G. Quantum Inverse Scattering Method and Correlation Functions 40–41 (Cambridge Univ. Press, Cambridge, 1993)

    Book  Google Scholar 

  28. Srednicki, M. Does quantum chaos explain quantum statistical mechanics? Preprint at 〈http://arXiv.org/abs/cond-mat/9410046〉 (1994)

  29. Srednicki, M. Thermal fluctuations in quantized chaotic systems. J. Phys. A 29, L75–L79 (1996)

    Article  CAS  ADS  Google Scholar 

  30. José, J. V. & Saletan, E. J. Classical Dynamics: A Contemporary Approach 474–491 (Cambridge Univ. Press, Cambridge, 1998)

    Book  Google Scholar 

Download references

Acknowledgements

We thank A. C. Cassidy, K. Jacobs, A. P. Young, and E. J. Heller for their comments. We acknowledge financial support from the National Science Foundation and the Office of Naval Research. We are grateful to the USC HPCC centre, where all our numerical computations were performed.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Maxim Olshanii.

Supplementary information

Supplementary Discussion

This file contains Supplementary Discussion of: 1. some details of the model and numerical calculations; 2. the considerations necessary when considering the microcanonical ensemble for a small system; and 3. the properties and the role of the width of the energy distribution. It contains Supplementary Figure 1-2 with Legends. (PDF 225 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008). https://doi.org/10.1038/nature06838

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature06838

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

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