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.

Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms


For a system at a temperature of absolute zero, all thermal fluctuations are frozen out, while quantum fluctuations prevail. These microscopic quantum fluctuations can induce a macroscopic phase transition in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value. Here we observe such a quantum phase transition in a Bose–Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. As the potential depth of the lattice is increased, a transition is observed from a superfluid to a Mott insulator phase. In the superfluid phase, each atom is spread out over the entire lattice, with long-range phase coherence. But in the insulating phase, exact numbers of atoms are localized at individual lattice sites, with no phase coherence across the lattice; this phase is characterized by a gap in the excitation spectrum. We can induce reversible changes between the two ground states of the system.

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: Schematic three-dimensional interference pattern with measured absorption images taken along two orthogonal directions.
Figure 2: Absorption images of multiple matter wave interference patterns.
Figure 3: Restoring coherence.
Figure 4: Excitation gap in the Mott insulator phase with exactly n = 1 atom per lattice site.
Figure 5: Probing the excitation probability versus an applied vertical potential gradient.
Figure 6: Energy difference between neighbouring lattice sites ΔE for which the Mott insulator phase can be resonantly perturbed versus the lattice potential depth Vmax.


  1. 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).

    Article  ADS  CAS  Google Scholar 

  2. 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).

    Article  ADS  CAS  Google Scholar 

  3. Stringari, S. Bose-Einstein condensation and superfluidity in trapped atomic gases. C.R. Acad. Sci. 4, 381–397 (2001).

    Google Scholar 

  4. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, Cambridge, 2001).

    MATH  Google Scholar 

  5. 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).

    Article  ADS  CAS  Google Scholar 

  6. Freericks, J. K. & Monien, H. Phase diagram of the Bose Hubbard model. Europhys. Lett. 26, 545–550 (1995).

    Article  ADS  Google Scholar 

  7. van Oosten, D., van der Straten, P. & Stoof, H. T. C. Quantum phases in an optical lattice. Phys. Rev. A 63, 053601-1–053601-12 (2001).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  9. Orr, B. G., Jaeger, H. M., Goldman, A. M. & Kuper, C. G. Global phase coherence in two-dimensional granular superconductors. Phys. Rev. Lett. 56, 378–381 (1986).

    Article  ADS  CAS  Google Scholar 

  10. Haviland, D. B., Liu, Y. & Goldman, A. M. Onset of superconductivity in the two-dimensional limit. Phys. Rev. Lett. 62, 2180–2183 (1989).

    Article  ADS  CAS  Google Scholar 

  11. Bradley, R. M. & Doniach, S. Quantum fluctuations in chains of Josephson junctions. Phys. Rev. B 30, 1138–1147 (1984).

    Article  ADS  Google Scholar 

  12. Geerligs, L. J., Peters, M., de Groot, L. E. M., Verbruggen, A. & Mooij, J. E. Charging effects and quantum coherence in regular Josephson junction arrays. Phys. Rev. Lett. 63, 326–329 (1989).

    Article  ADS  CAS  Google Scholar 

  13. Zwerger, W. Global and local phase coherence in dissipative Josephson-junction arrays. Europhys. Lett. 9, 421–426 (1989).

    Article  ADS  Google Scholar 

  14. van der Zant, H. S. J., Fritschy, F. C., Elion, W. J., Geerligs, L. J. & Mooij, J. E. Field-induced superconductor-to-insulator transitions in Josephson-junction arrays. Phys. Rev. Lett. 69, 2971–2974 (1992).

    Article  ADS  CAS  Google Scholar 

  15. van Oudenaarden, A. & Mooij, J. E. One-dimensional Mott insulator formed by quantum vortices in Josephson junction arrays. Phys. Rev. Lett. 76, 4947–4950 (1996).

    Article  ADS  CAS  Google Scholar 

  16. Chow, E., Delsing, P. & Haviland, D. B. Length-scale dependence of the superconductor-to-insulator quantum phase transition in one dimension. Phys. Rev. Lett. 81, 204–207 (1998).

    Article  ADS  CAS  Google Scholar 

  17. Orzel, C., Tuchman, A. K., Fenselau, M. L., Yasuda, M. & Kasevich, M. A. Squeezed states in a Bose-Einstein condensate. Science 291, 2386–2389 (2001).

    Article  ADS  CAS  Google Scholar 

  18. Greiner, M., Bloch, I., Hänsch, T. W. & Esslinger, T. Magnetic transport of trapped cold atoms over a large distance. Phys. Rev. A 63, 031401-1–031401-4 (2001).

    Article  ADS  Google Scholar 

  19. Greiner, M., Bloch, I., Mandel, O., Hänsch, T. W. & Esslinger, T. Exploring phase coherence in a 2D lattice of Bose-Einstein condensates. Phys. Rev. Lett. 87, 160405-1–160405-4 (2001).

    ADS  Google Scholar 

  20. Grimm, R., Weidemüller, M. & Ovchinnikov, Yu. B. Optical dipole traps for neutral atoms. Adv. At. Mol. Opt. Phys. 42, 95–170 (2000).

    Article  ADS  CAS  Google Scholar 

  21. Kastberg, A., Phillips, W. D., Rolston, S. L., Spreeuw, R. J. C. & Jessen, P. S. Adiabatic cooling of cesium to 700 nK in an optical lattice. Phys. Rev. Lett. 74, 1542–1545 (1995).

    Article  ADS  CAS  Google Scholar 

  22. Dalfovo, F. D., Giorgini, S., Pitaevskii, L. P. & Stringari, S. Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999).

    Article  ADS  CAS  Google Scholar 

  23. Inouye, S. et al. Observation of Feshbach resonances in a Bose–Einstein condensate. Nature 392, 151–154 (1998).

    Article  ADS  CAS  Google Scholar 

  24. Donley, E. A. et al. Dynamics of collapsing and exploding Bose–Einstein condensates. Nature 412, 295–299 (2001).

    Article  ADS  CAS  Google Scholar 

  25. Bouyer, P. & Kasevich, M. Heisenberg-limited spectroscopy with degenerate Bose-Einstein gases. Phys. Rev. A 56, R1083–R1086 (1997).

    Article  ADS  CAS  Google Scholar 

  26. Jaksch, D., Briegel, H.-J., Cirac, J. I., Gardiner, C. W. & Zoller, P. Entanglement of atoms via cold controlled collisions. Phys. Rev. Lett. 82, 1975–1978 (1999).

    Article  ADS  CAS  Google Scholar 

Download references


We thank W. Zwerger, H. Monien, I. Cirac, K. Burnett and Yu. Kagan for discussions. This work was supported by the DFG, and by the EU under the QUEST programme.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Immanuel Bloch.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Greiner, M., Mandel, O., Esslinger, T. et al. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI:

This article is cited by


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.


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