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.

Suppression of the critical temperature for superfluidity near the Mott transition


Ultracold atomic gases in optical lattices have proven to be a controllable, tunable and clean implementation of strongly interacting quantum many-body systems. An essential prospect for such quantum simulators is their ability to map out the phase diagram of fundamental many-body model Hamiltonians. However, the results need to be validated first for representative benchmark problems through state-of-the-art numerical methods of quantum many-body theory. Here we present the first ab initio comparison between experiments and quantum Monte Carlo simulations for strongly interacting Bose gases on a lattice for large systems (up to particles). The comparison enables thermometry for the interacting quantum gas and to experimentally determine the finite-temperature phase diagram for bosonic superfluids in an optical lattice, revealing a suppression of the critical temperature as the transition to the Mott insulator is approached.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Simplified scheme of the finite T phase diagram for a single species of bosons in a lattice potential at density n=1.
Figure 2: Thermodynamic properties of the Bose–Hubbard model at finite temperatures.
Figure 3: Comparison of experimental and simulated TOF distributions.
Figure 4: Fit results for the onset of superfluidity.
Figure 5: Finite temperature phase diagram and suppression of Tc in the lattice.


  1. 1

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

    ADS  Article  Google Scholar 

  2. 2

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

    ADS  Article  Google Scholar 

  3. 3

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

    ADS  Article  Google Scholar 

  4. 4

    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).

    ADS  Article  Google Scholar 

  5. 5

    Paredes, B. et al. Tonks–Girardeau gas of ultracold atoms in an optical lattice. Nature 429, 277–281 (2004).

    ADS  Article  Google Scholar 

  6. 6

    Kinoshita, T., R. W., T. & Weiss, D. S. Observation of a one-dimensional Tonks–Girardeau gas. Science 305, 1125–1128 (2004).

    ADS  Article  Google Scholar 

  7. 7

    Hadzibabic, Z., Krüger, P., Cheneau, M., Battelier, B. & Dalibard, J. Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas. Nature 441, 1118–1121 (2006).

    ADS  Article  Google Scholar 

  8. 8

    Fisher, M. P. A., Weichman, P. B., Grinstein, G. & Fisher, D. S. Boson localization and the superfluid–insulator transition. Phys. Rev. B 40, 546–570 (1989).

    ADS  Article  Google Scholar 

  9. 9

    Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108–3111 (1998).

    ADS  Article  Google Scholar 

  10. 10

    Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, 1999).

    MATH  Google Scholar 

  11. 11

    Stöferle, T., Moritz, H., Schori, C., Köhl, M. & Esslinger, T. Transition form a strongly interacting 1d superfluid to a Mott insulator. Phys. Rev. Lett. 92, 130403 (2004).

    ADS  Article  Google Scholar 

  12. 12

    Gerbier, F. et al. Phase coherence of an atomic Mott insulator. Phys. Rev. Lett. 95, 050404 (2005).

    ADS  Article  Google Scholar 

  13. 13

    Gerbier, F., Fölling, S., Widera, A., Mandel, O. & Bloch, I. Probing number squeezing of ultracold atoms across the superfluid-Mott insulator transition. Phys. Rev. Lett. 96, 090401 (2006).

    ADS  Article  Google Scholar 

  14. 14

    Fölling, S., Widera, A., Müller, T., Gerbier, F. & Bloch, I. Formation of spatial shell structure in the superfluid to Mott insulator transition. Phys. Rev. Lett. 97, 060403 (2006).

    ADS  Article  Google Scholar 

  15. 15

    Campbell, G. K. et al. Imaging the Mott insulator shells by using atomic clock shifts. Science 313, 649–652 (2006).

    ADS  Article  Google Scholar 

  16. 16

    Spielman, I. B., Phillips, W. D. & Porto, J. V. Mott-insulator transition in a two-dimensional atomic Bose gas. Phys. Rev. Lett. 98, 080404 (2007).

    ADS  Article  Google Scholar 

  17. 17

    Mun, J., Campbell, G. K., Marcassa, L. G., Pritchard, D. E. & Ketterle, W. Phase diagram for a Bose–Einstein condensate moving in an optical lattice. Phys. Rev. Lett. 99, 150604 (2007).

    ADS  Article  Google Scholar 

  18. 18

    Spielman, I. B., Phillips, W. D. & Porto, J. V. Condensate fraction in a 2d Bose gas measured across the Mott-insulator transition. Phys. Rev. Lett. 100, 120402 (2008).

    ADS  Article  Google Scholar 

  19. 19

    Guarrera, V. et al. Noise correlation spectroscopy of the broken order of a Mott insulating phase. Phys. Rev. Lett. 100, 250403 (2008).

    ADS  Article  Google Scholar 

  20. 20

    Sheshadri, K., Krishnamurthy, H. R., Pandit, R. & Ramakrishnan, T. V. Superfluid and insulating phases in an interacting-boson model: Mean-field theory and the RPA. Europhys. Lett. 22, 257–263 (1993).

    ADS  Article  Google Scholar 

  21. 21

    Elstner, N. & Monien, H. Dynamics and thermodynamics of the Bose–Hubbard model. Phys. Rev. B 59, 12184–12187 (1999).

    ADS  Article  Google Scholar 

  22. 22

    Dickerscheid, D. B. M., van Oosten, D., Denteneer, P. J. H. & Stoof, H. T. C. Ultracold atoms in optical lattices. Phys. Rev. A 68, 043623 (2003).

    ADS  Article  Google Scholar 

  23. 23

    DeMarco, B., Lannert, C., Vishveshwara, S. & Wei, T-C. Structure and stability of Mott-insulator shells of bosons trapped in an optical lattice. Phys. Rev. A 71, 063601 (2005).

    ADS  Article  Google Scholar 

  24. 24

    Pupillo, G., Williams, C. J. & Prokof’ev, N. V. Effects of finite temperature on the Mott-insulator state. Phys. Rev. A 73, 013408 (2006).

    ADS  Article  Google Scholar 

  25. 25

    Blakie, P. B., Rey, A-M. & Bezett, A. Thermodynamics of quantum degenerate gases in optical lattices. Laser Phys. 17, 198–204 (2007).

    ADS  Article  Google Scholar 

  26. 26

    Capogrosso-Sansone, B., Prokof’ev, N. & Svistunov, B. Phase diagram and thermodynamics of the three-dimensional Bose–Hubbard model. Phys. Rev. B 75, 134302 (2007).

    ADS  Article  Google Scholar 

  27. 27

    Diener, R. B., Zhou, Q., Zhai, H. & Ho, T-L. Criterion for bosonic superfluidity in an optical lattice. Phys. Rev. Lett. 98, 180404 (2007).

    ADS  Article  Google Scholar 

  28. 28

    Kato, Y., Zhou, Q., Kawashima, N. & Trivedi, N. Sharp peaks in the momentum distribution of bosons in optical lattices in the normal state. Nature Phys. 4, 617–621 (2008).

    ADS  Article  Google Scholar 

  29. 29

    Gerbier, F. et al. Expansion of a quantum gas released from an optical lattice. Phys. Rev. Lett. 101, 155303 (2008).

    ADS  Article  Google Scholar 

  30. 30

    Kollath, C., Schollwöck, U., von Delft, J. & Zwerger, W. Spatial correlations of trapped one-dimensional bosons in an optical lattice. Phys. Rev. A 69, 031601 (2004).

    ADS  Article  Google Scholar 

  31. 31

    Kashurnikov, V., Prokofiev, N. & Svistunov, B. Revealing the superfluid–Mott-insulator transition in an optical lattice. Phys. Rev. A 66, 031601 (2002).

    ADS  Article  Google Scholar 

  32. 32

    Muradyan, G. & Anglin, J. R. Finite-temperature coherence of the ideal Bose gas in an optical lattice. Phys. Rev. A 78, 053628 (2008).

    ADS  Article  Google Scholar 

  33. 33

    McKay, D., White, M. & DeMarco, B. Lattice thermodynamics for ultracold atoms. Phys. Rev. A 79, 063605 (2009).

    ADS  Article  Google Scholar 

  34. 34

    Rey, A. M., Pupillo, G. & Porto, J. V. The role of interactions, tunneling and harmonic confinement on the adiabatic loading of bosons in an optical lattice. Phys. Rev. A 73, 023608 (2006).

    ADS  Article  Google Scholar 

  35. 35

    Ho, T-L. & Zhou, Q. Intrinsic heating and cooling in adiabatic processes for bosons in optical lattices. Phys. Rev. Lett. 99, 120404 (2007).

    ADS  Article  Google Scholar 

  36. 36

    Gerbier, F. Boson Mott insulators at finite temperatures. Phys. Rev. Lett. 99, 120405 (2007).

    ADS  Article  Google Scholar 

  37. 37

    Pollet, L., Kollath, C., Houcke, K. V. & Troyer, M. Temperature changes when adiabatically ramping up an optical lattice. New J. Phys. 10, 065001 (2008).

    ADS  Article  Google Scholar 

  38. 38

    Zhou, Q., Kato, Y., Kawashima, N. & Trivedi, N. Direct mapping of the finite temperature phase diagram of strongly correlated quantum models. Phys. Rev. Lett. 103, 085701 (2009).

    ADS  Article  Google Scholar 

  39. 39

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

    ADS  Article  Google Scholar 

  40. 40

    Prokof’ev, N. V., Svistunov, B. V. & Tupitsyn, I. Exact, complete, and universal continuous-time worldline Monte Carlo approach to the statistics of discrete quantum systems. Sov. Phys. JETP 87, 310–321 (1998).

    ADS  Article  Google Scholar 

  41. 41

    Pollet, L., Houcke, K. V. & Rombouts, S. M. A. Engineering local optimality in quantum Monte Carlo algorithms. J. Comput. Phys. 225, 2249–2266 (2007).

    ADS  Article  Google Scholar 

  42. 42

    Fölling, S. et al. Spatial quantum noise interferometry in expanding ultracold atom clouds. Nature 434, 481–484 (2005).

    ADS  Article  Google Scholar 

  43. 43

    Leggett, A. J. Superfluidity. Rev. Mod. Phys. 71, S318–S323 (1999).

    Article  Google Scholar 

  44. 44

    Gerbier, F. et al. Critical temperature of a trapped, weakly interacting Bose gas. Phys. Rev. Lett. 92, 030405 (2004).

    ADS  Article  Google Scholar 

  45. 45

    Ketterle, W., Durfee, D. S. & Stamper-Kurn, D. M. in Proc. Int. School of Physics—Enrico Fermi (eds Inguscio, M., Stringari, S. & Wieman, C. E.) 67–176 (IOS Press, 1999).

    Google Scholar 

Download references


We would like to thank B. Capogrosso-Sansone, P. N. Ma, F. C. Zhang, S. Fölling, H. Moritz, T. Esslinger and J. Dalibard for stimulating discussions. This work was supported by the DFG, the SNF, the EU (IP SCALA), DARPA (OLE program) and AFOSR. The simulations were run on the Brutus cluster at the ETH in Zürich.

Author information




S.T. and U.S. carried out the experiments and L.P., B.S., M.T. and N.V.P. performed the numerical simulations. S.T., L.P., F.G., N.V.P., B.S. and M.T. were involved in the analysis of the data. All authors conceived the research, discussed the results and wrote the manuscript.

Corresponding author

Correspondence to S. Trotzky.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 415 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Trotzky, S., Pollet, L., Gerbier, F. et al. Suppression of the critical temperature for superfluidity near the Mott transition. Nature Phys 6, 998–1004 (2010).

Download citation

Further reading


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