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Entanglement pre-thermalization in a one-dimensional Bose gas

Nature Physics volume 11, pages 1050–1056 (2015)Cite this article

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Abstract

An isolated quantum system often shows relaxation to a quasi-stationary state before reaching thermal equilibrium. Such a pre-thermalized state was observed in recent experiments in a one-dimensional Bose gas after it had been coherently split into two. Although the existence of local conserved quantities is usually considered to be the key ingredient of pre-thermalization, the question of whether non-local correlations between the subsystems can influence pre-thermalization of the entire system has remained unanswered. Here we study the dynamics of coherently split one-dimensional Bose gases and find that the initial entanglement combined with energy degeneracy due to parity and translation invariance strongly affects the long-term behaviour of the system. The mechanism of this entanglement pre-thermalization is quite general and not restricted to one-dimensional Bose gases. In view of recent experiments with a small and well-defined number of ultracold atoms, our predictions based on exact few-body calculations could be tested in experiments.

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Figure 1: Schematic illustration of our set-up and time-averaged auto- and cross-correlation functions in the course of pre-thermalization.
Figure 2: Comparison of the time-averaged auto-correlation functions.
Figure 3: Pre-thermalization and entanglement pre-thermalization exhibited by two-point correlation functions.
Figure 4: The system-size dependence of the deviation of the auto-correlation (red) and cross-correlation (blue) functions from the equilibrium ones.
Figure 5: Diagonal (purple dashed) and off-diagonal (green dashed) contributions in the infinite-time average of the cross-correlation function for N = 3.

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References

  1. Neumann, J. v. Beweis des ergodensatzes und des H-theorems in der neuen mechanik. Z. Phys. 57, 30–70 (1929).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  3. Tasaki, H. From quantum dynamics to the canonical distribution: General picture and a rigorous example. Phys. Rev. Lett. 80, 1373–1376 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  4. Popescu, S., Short, A. J. & Winter, A. Entanglement and the foundations of statistical mechanics. Nature Phys. 2, 754–758 (2006).

    Article  ADS  Google Scholar 

  5. Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).

    Article  ADS  Google Scholar 

  6. Linden, N., Popescu, S., Short, A. J. & Winter, A. Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  7. Goldstein, S., Lebowitz, J. L., Mastrodonato, C., Tumulka, R. & Zanghì, N. Normal typicality and von Neumann’s quantum ergodic theorem. Proc. R. Soc. A 466, 3203–3224 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  8. Sato, J., Kanamoto, R., Kaminishi, E. & Deguchi, T. Exact relaxation dynamics of a localized many-body state in the 1D Bose gas. Phys. Rev. Lett. 108, 110401 (2012).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. 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  ADS  Google Scholar 

  11. Trotzky, S. et al. Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas. Nature Phys. 8, 325–330 (2012).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  13. Gring, M. et al. Relaxation and prethermalization in an isolated quantum system. Science 337, 1318–1322 (2012).

    Article  ADS  Google Scholar 

  14. Langen, T., Geiger, R., Kuhnert, M., Rauer, B. & Schmiedmayer, J. Local emergence of thermal correlations in an isolated quantum many-body system. Nature Phys. 9, 640–643 (2013).

    Article  ADS  Google Scholar 

  15. Kollar, M., Wolf, F. A. & Eckstein, M. Generalized Gibbs ensemble prediction of prethermalization plateaus and their relation to nonthermal steady states in integrable systems. Phys. Rev. B 84, 054304 (2011).

    Article  ADS  Google Scholar 

  16. van den Worm, M., Sawyer, B. C., Bollinger, J. J. & Kastner, M. Relaxation timescales and decay of correlations in a long-range interacting quantum simulator. New J. Phys. 15, 083007 (2013).

    Article  ADS  Google Scholar 

  17. Eckstein, M., Kollar, M. & Werner, P. Thermalization after an interaction quench in the Hubbard model. Phys. Rev. Lett. 103, 056403 (2009).

    Article  ADS  Google Scholar 

  18. Iyer, D. & Andrei, N. Quench dynamics of the interacting Bose gas in one dimension. Phys. Rev. Lett. 109, 115304 (2012).

    Article  ADS  Google Scholar 

  19. Kormos, M., Shashi, A., Chou, Y.-Z., Caux, J.-S. & Imambekov, A. Interaction quenches in the one-dimensional Bose gas. Phys. Rev. B 88, 205131 (2013).

    Article  ADS  Google Scholar 

  20. Kormos, M., Collura, M. & Calabrese, P. Analytic results for a quantum quench from free to hard-core one-dimensional Bosons. Phys. Rev. A 89, 013609 (2014).

    Article  ADS  Google Scholar 

  21. De Nardis, J., Wouters, B., Brockmann, M. & Caux, J.-S. Solution for an interaction quench in the Lieb–Liniger Bose gas. Phys. Rev. A 89, 033601 (2014).

    Article  ADS  Google Scholar 

  22. Goldstein, G. & Andrei, N. Quench between a Mott insulator and a Lieb–Liniger liquid. Phys. Rev. A 90, 043626 (2014).

    Article  ADS  Google Scholar 

  23. Rossini, D., Silva, A., Mussardo, G. & Santoro, G. E. Effective thermal dynamics following a quantum quench in a spin chain. Phys. Rev. Lett. 102, 127204 (2009).

    Article  ADS  Google Scholar 

  24. Rossini, D., Suzuki, S., Mussardo, G., Santoro, G. E. & Silva, A. Long time dynamics following a quench in an integrable quantum spin chain: Local versus nonlocal operators and effective thermal behavior. Phys. Rev. B 82, 144302 (2010).

    Article  ADS  Google Scholar 

  25. Calabrese, P., Essler, F. H. L. & Fagotti, M. Quantum quench in the transverse-field Ising chain. Phys. Rev. Lett. 106, 227203 (2011).

    Article  ADS  Google Scholar 

  26. Calabrese, P., Essler, F. H. & Fagotti, M. Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators. J. Stat. Mech. 2012, P07016 (2012).

    Google Scholar 

  27. Calabrese, P., Essler, F. H. & Fagotti, M. Quantum quenches in the transverse field Ising chain: II. Stationary state properties. J. Stat. Mech. 2012, P07022 (2012).

    Google Scholar 

  28. 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 

  29. Cassidy, A. C., Clark, C. W. & Rigol, M. Generalized thermalization in an integrable lattice system. Phys. Rev. Lett. 106, 140405 (2011).

    Article  ADS  Google Scholar 

  30. Mussardo, G. Infinite-time average of local fields in an integrable quantum field theory after a quantum quench. Phys. Rev. Lett. 111, 100401 (2013).

    Article  ADS  Google Scholar 

  31. Sotiriadis, S. & Calabrese, P. Validity of the GGE for quantum quenches from interacting to noninteracting models. J. Stat. Mech. 2014, P07024 (2014).

    Article  MathSciNet  Google Scholar 

  32. Essler, F. H. L., Mussardo, G. & Panfil, M. Generalized Gibbs ensembles for quantum field theories. Phys. Rev. A 91, 051602(R) (2015).

    Article  ADS  Google Scholar 

  33. Serwane, F. et al. Deterministic preparation of a tunable few-fermion system. Science 332, 336–338 (2011).

    Article  ADS  Google Scholar 

  34. Zürn, G. et al. Fermionization of two distinguishable fermions. Phys. Rev. Lett. 108, 075303 (2012).

    Article  ADS  Google Scholar 

  35. Kitagawa, T. et al. Ramsey interference in one-dimensional systems: The full distribution function of fringe contrast as a probe of many-body dynamics. Phys. Rev. Lett. 104, 255302 (2010).

    Article  ADS  Google Scholar 

  36. Kitagawa, T., Imambekov, A., Schmiedmayer, J. & Demler, E. The dynamics and prethermalization of one-dimensional quantum systems probed through the full distributions of quantum noise. New J. Phys. 13, 073018 (2011).

    Article  ADS  Google Scholar 

  37. Lieb, E. H. & Liniger, W. Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963).

    Article  ADS  MathSciNet  Google Scholar 

  38. Caux, J.-S., Calabrese, P. & Slavnov, N. A. One-particle dynamical correlations in the one-dimensional Bose gas. J. Stat. Mech. 2007, P01008 (2007).

    Article  Google Scholar 

  39. Gaudin, M. La fonction d’onde de Bethe (Masson, 1983).

    MATH  Google Scholar 

  40. Korepin, V. Calculation of norms of Bethe wave functions. Commun. Math. Phys. 86, 391–418 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  41. Slavnov, N. A. Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe ansatz. Teor. Mat. Fiz. 79, 502–508 (1989).

    Article  MathSciNet  Google Scholar 

  42. Slavnov, N. A. Nonequal-time current correlation function in a one-dimensional Bose gas. Teor. Mat. Fiz. 82, 273–282 (1990).

    Article  Google Scholar 

  43. Reimann, P. Foundation of statistical mechanics under experimentally realistic conditions. Phys. Rev. Lett. 101, 190403 (2008).

    Article  ADS  Google Scholar 

  44. Short, A. J. Equilibration of quantum systems and subsystems. New J. Phys. 13, 053009 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  45. Fagotti, M. & Essler, F. H. L. Reduced density matrix after a quantum quench. Phys. Rev. B 87, 245107 (2013).

    Article  ADS  Google Scholar 

  46. Olshanii, M. Atomic scattering in the presence of an external confinement and a gas of impenetrable Bosons. Phys. Rev. Lett. 81, 938–941 (1998).

    Article  ADS  Google Scholar 

  47. Rigol, M. Breakdown of thermalization in finite one-dimensional systems. Phys. Rev. Lett. 103, 100403 (2009).

    Article  ADS  Google Scholar 

  48. Caux, J.-S. & Essler, F. H. L. Time evolution of local observables after quenching to an integrable model. Phys. Rev. Lett. 110, 257203 (2013).

    Article  ADS  Google Scholar 

  49. Pozsgay, B. et al. Correlations after quantum quenches in the XXZ spin chain: Failure of the generalized Gibbs ensemble. Phys. Rev. Lett. 113, 117203 (2014).

    Article  ADS  Google Scholar 

  50. De Nardis, J., Piroli, L. & Caux, J.-S. Relaxation dynamics of local observables in integrable systems. Preprint at http://arXiv.org/abs/1505.03080 (2015).

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Acknowledgements

We thank T. Deguchi, S. Furukawa and N. Sakumichi for discussions. This work was supported by KAKENHI Grant No. 26287088 from the Japan Society for the Promotion of Science, a Grant-in-Aid for Scientific Research on Innovative Areas ‘Topological Materials Science’ (KAKENHI Grant No. 15H05855), the Photon Frontier Network Program from MEXT of Japan, and the Mitsubishi Foundation. E.K. acknowledges support from the Institute for Photon Science and Technology. T.M. acknowledges the JSPS Core-to-Core Program ‘Non-equilibrium dynamics of soft matter and information’ for financial support. T.N.I. acknowledges the JSPS for financial support (Grant No. 248408) and Postdoctoral Fellowship for Research Abroad.

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Authors and Affiliations

  1. Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

    Eriko Kaminishi, Takashi Mori, Tatsuhiko N. Ikeda & Masahito Ueda

  2. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

    Tatsuhiko N. Ikeda

  3. RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

    Masahito Ueda

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  1. Eriko Kaminishi
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All authors contributed extensively to the work presented in the paper and the writing of the manuscript.

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Correspondence to Eriko Kaminishi.

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Kaminishi, E., Mori, T., Ikeda, T. et al. Entanglement pre-thermalization in a one-dimensional Bose gas. Nature Phys 11, 1050–1056 (2015). https://doi.org/10.1038/nphys3478

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