Skip to main content

Thank you for visiting 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.

Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms


Transport properties are among the defining characteristics of many important phases in condensed-matter physics. In the presence of strong correlations they are difficult to predict, even for model systems such as the Hubbard model. In real materials, additional complications arise owing to impurities, lattice defects or multi-band effects. Ultracold atoms in contrast offer the possibility to study transport and out-of-equilibrium phenomena in a clean and well-controlled environment and can therefore act as a quantum simulator for condensed-matter systems. Here we studied the expansion of an initially confined fermionic quantum gas in the lowest band of a homogeneous optical lattice. For non-interacting atoms, we observe ballistic transport, but even small interactions render the expansion almost bimodal, with a dramatically reduced expansion velocity. The dynamics is independent of the sign of the interaction, revealing a novel, dynamic symmetry of the Hubbard model.

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

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Expansion of fermionic atoms after a quench of the trapping potential.
Figure 2: Expansion of non-interacting fermions.
Figure 3: Expansion of interacting fermions.
Figure 4: Core-expansion velocities.


  1. Jaksch, D. & Zoller, P. The cold atom Hubbard toolbox. Ann. Phys. 315, 52–79 (2005).

    Article  ADS  Google Scholar 

  2. Lewenstein, M. et al. Ultracold atomic gases in optical lattices: Mimicking condensed matter physics and beyond. Adv. Phys. 56, 243–379 (2007).

    Article  ADS  Google Scholar 

  3. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).

    Article  ADS  Google Scholar 

  4. Hung, C., Zhang, X., Gemelke, N. & Chin, C. Slow mass transport and statistical evolution of an atomic gas across the superfluid-Mott-insulator transition. Phys. Rev. Lett. 104, 160403 (2010).

    Article  ADS  Google Scholar 

  5. Strohmaier, N. et al. Observation of elastic doublon decay in the Fermi–Hubbard model. Phys. Rev. Lett. 104, 080401 (2010).

    Article  ADS  Google Scholar 

  6. Wernsdorfer, J., Snoek, M. & Hofstetter, W. Lattice-ramp-induced dynamics in an interacting Bose–Bose mixture. Phys. Rev. A 81, 043620 (2010).

    Article  ADS  Google Scholar 

  7. Gericke, T. et al. Adiabatic loading of a Bose–Einstein condensate in a 3D optical lattice. J. Mod. Opt. 54, 735–743 (2007).

    Article  ADS  Google Scholar 

  8. Hackermüller, L. et al. Anomalous expansion of attractively interacting fermionic atoms in an optical lattice. Science 327, 1621–1624 (2010).

    Article  ADS  Google Scholar 

  9. Jördens, R., Strohmaier, N., Günter, K., Moritz, H. & Esslinger, T. A Mott insulator of fermionic atoms in an optical lattice. Nature 455, 204–207 (2008).

    Article  ADS  Google Scholar 

  10. Schneider, U. et al. Metallic and insulating phases of repulsively interacting fermions in a 3D optical lattice. Science 322, 1520–1525 (2008).

    Article  ADS  Google Scholar 

  11. Hubbard, J. Electron correlations in narrow energy bands. Proc. R. Soc. A 276, 238–257 (1963).

    ADS  Google Scholar 

  12. Pezzè, L. et al. Insulating behavior of a trapped ideal Fermi gas. Phys. Rev. Lett. 93, 120401 (2004).

    Article  ADS  Google Scholar 

  13. Ott, H. et al. Collisionally induced transport in periodic potentials. Phys. Rev. Lett. 92, 160601 (2004).

    Article  ADS  Google Scholar 

  14. Strohmaier, N. et al. Interaction-controlled transport of an ultracold Fermi gas. Phys. Rev. Lett. 99, 220601 (2007).

    Article  ADS  Google Scholar 

  15. Lignier, H. et al. Dynamical control of matter-wave tunneling in periodic potentials. Phys. Rev. Lett. 99, 220403 (2007).

    Article  ADS  Google Scholar 

  16. Ben Dahan, M., Peik, E., Reichel, J., Castin, Y. & Salomon, C. Bloch oscillations of atoms in an optical potential. Phys. Rev. Lett. 76, 4508–4511 (1996).

    Article  ADS  Google Scholar 

  17. Fertig, C. D. et al. Strongly inhibited transport of a degenerate 1D Bose gas in a lattice. Phys. Rev. Lett. 94, 120403 (2005).

    Article  ADS  Google Scholar 

  18. Gustavsson, M. et al. Control of interaction-induced dephasing of Bloch oscillations. Phys. Rev. Lett. 100, 080404 (2008).

    Article  ADS  Google Scholar 

  19. Fattori, M. et al. Atom interferometry with a weakly interacting Bose–Einstein condensate. Phys. Rev. Lett. 100, 080405 (2008).

    Article  ADS  Google Scholar 

  20. Aharonov, Y., Davidovich, L. & Zagury, N. Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993).

    Article  ADS  Google Scholar 

  21. Farhi, E. & Gutmann, S. Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  22. Karski, M. et al. Quantum walk in position space with single optically trapped atoms. Science 325, 174–177 (2009).

    Article  ADS  Google Scholar 

  23. Weitenberg, C. et al. Single-spin addressing in an atomic Mott insulator. Nature 471, 319–324 (2011).

    Article  ADS  Google Scholar 

  24. Childs, A. M. et al. in STOC ’03: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing 59–68 (2003).

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

    Article  ADS  Google Scholar 

  26. 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 

  27. Mandt, S., Rapp, A. & Rosch, A. Interacting fermionic atoms in optical lattices diffuse symmetrically upwards and downwards in a gravitational potential. Phys. Rev. Lett. 106, 250602 (2011).

    Article  ADS  Google Scholar 

  28. Vázquez, J. L. Smoothing and Decay Estimates for Nonlinear Diffusion Equations (Oxford Univ. Press, 2006).

    Book  Google Scholar 

  29. Anker, Th. et al. Nonlinear self-trapping of matter waves in periodic potentials. Phys. Rev. Lett. 94, 020403 (2005).

    Article  ADS  Google Scholar 

  30. Romero-Isart, O., Eckert, K., Rodo, C. & Sanpera, A. Transport and entanglement generation in the Bose–Hubbard model. J. Phys. A 40, 8019–8031 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  31. Kajala, J., Massel, J. & Törmä, P. Expansion dynamics in the one-dimensional Fermi–Hubbard model. Phys. Rev. Lett. 106, 206401 (2011).

    Article  ADS  Google Scholar 

  32. Schollwöck, U. The density matrix renormalization group. Rev. Mod. Phys. 77, 259–315 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  33. Heidrich-Meisner, F. et al. Quantum distillation: Dynamical generation of low-entropy states of strongly correlated fermions in an optical lattice. Phys. Rev. A 80, 041603 (2009).

    Article  ADS  Google Scholar 

  34. Rapp, A., Mandt, S. & Rosch, A. Equilibration rates and negative absolute temperatures for ultracold atoms in optical lattices. Phys. Rev. Lett. 105, 220405 (2010).

    Article  ADS  Google Scholar 

  35. Purcell, E. M. & Pound, R. V. A nuclear spin system at negative temperature. Phys. Rev. 81, 279–280 (1951).

    Article  ADS  Google Scholar 

  36. Medley, P., Weld, D. M., Miyake, H., Pritchard, D. E. & Ketterle, W. Spin gradient demagnetization cooling of ultracold atoms. Phys. Rev. Lett. 106, 195301 (2011).

    Article  ADS  Google Scholar 

  37. McKay, D. C. & DeMarco, B. Cooling in strongly correlated optical lattices: Prospects and challenges. Rep. Prog. Phys. 74, 054401 (2011).

    Article  ADS  Google Scholar 

Download references


We thank M. Moreno-Cardoner, F. Heidrich-Meisner, D. Pekker and R. Sensarma, B. Kawohl, C. Kiefer, J. Krug and M. Zirnbauer for stimulating and insightful discussions.

This work was supported by the Deutsche Forschungsgemeinschaft (FOR801, SFB TR 12, SFB 608, Gottfried Wilhelm Leibniz Prize), the European Union (Integrated Project SCALA), EuroQUAM (L.H.), the US Defense Advanced Research Projects Agency (Optical Lattice Emulator program), the US Air Force Office of Scientific Research (Quantum Simulation MURI (E.D.)), the National Science Foundation (DMR-07-05472) (E.D.), the Harvard-MIT CUA (E.D.), MATCOR (S.W.), the Gutenberg Akademie (S.W.) and the German National Academic Foundation (S.M.).

Author information

Authors and Affiliations



U.S., L.H. and J.P.R. carried out the measurements, U.S. performed the data analysis with contributions from L.H. and J.P.R.. I.B. supervised the measurements. S.M. and D.R. performed the numerical calculations supervised by A.R.. E.D., U.S. and A.R. constructed the analytical proof of the dynamical symmetry. U.S. and A.R. wrote the manuscript with substantial contributions by I.B. and all authors.

Corresponding author

Correspondence to Ulrich Schneider.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 1172 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Schneider, U., Hackermüller, L., Ronzheimer, J. et al. Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms. Nature Phys 8, 213–218 (2012).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


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