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Tonks–Girardeau gas of ultracold atoms in an optical lattice


Strongly correlated quantum systems are among the most intriguing and fundamental systems in physics. One such example is the Tonks–Girardeau gas1,2, proposed about 40 years ago, but until now lacking experimental realization; in such a gas, the repulsive interactions between bosonic particles confined to one dimension dominate the physics of the system. In order to minimize their mutual repulsion, the bosons are prevented from occupying the same position in space. This mimics the Pauli exclusion principle for fermions, causing the bosonic particles to exhibit fermionic properties1,2. However, such bosons do not exhibit completely ideal fermionic (or bosonic) quantum behaviour; for example, this is reflected in their characteristic momentum distribution3. Here we report the preparation of a Tonks–Girardeau gas of ultracold rubidium atoms held in a two-dimensional optical lattice formed by two orthogonal standing waves. The addition of a third, shallower lattice potential along the long axis of the quantum gases allows us to enter the Tonks–Girardeau regime by increasing the atoms' effective mass and thereby enhancing the role of interactions. We make a theoretical prediction of the momentum distribution based on an approach in which trapped bosons acquire fermionic properties, finding that it agrees closely with the measured distribution.

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Figure 1: Experimental sequence and momentum profiles.
Figure 2: Momentum profiles of the 1D quantum gases for different axial lattice depths.
Figure 3: Momentum profiles of a single 1D tube obtained from our fermionization-based theory for different lattice depths.


  1. Girardeau, M. Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. Phys. 1, 516–523 (1960)

    ADS  MathSciNet  Article  Google Scholar 

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

    ADS  MathSciNet  Article  Google Scholar 

  3. Lenard, A. Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons. J. Math. Phys. 5, 930–943 (1964)

    ADS  MathSciNet  Article  Google Scholar 

  4. Petrov, D. S., Shlyapnikov, G. V. & Walraven, J. T. M. Regimes of quantum degeneracy in trapped 1D gases. Phys. Rev. Lett. 85, 3745–3749 (2000)

    ADS  CAS  Article  Google Scholar 

  5. Dunjko, V., Lorent, V. & Olshanii, M. Bosons in cigar-shaped traps: Thomas-Fermi regime, Tonks-Girardeau regime, and in between. Phys. Rev. Lett. 86, 5413–5416 (2001)

    ADS  CAS  Article  Google Scholar 

  6. Jochim, S. et al. Bose-Einstein condensation of molecules. Science 302, 2101–2103 (2003)

    ADS  CAS  Article  Google Scholar 

  7. Greiner, M., Regal, C. & Jin, D. S. Emergence of a molecular Bose-Einstein condensate from a Fermi gas. Nature 426, 537–540 (2003)

    ADS  CAS  Article  Google Scholar 

  8. Zwierlein, M. W. et al. Observation of Bose-Einstein condensation of molecules. Phys. Rev. Lett. 91, 250401 (2003)

    ADS  CAS  Article  Google Scholar 

  9. Regal, C., Greiner, M. & Jin, D. S. Observation of resonance condensation of fermionic atom pairs. Phys. Rev. Lett. 92, 040403 (2004)

    ADS  CAS  Article  Google Scholar 

  10. Olshanii, M. Atomic scattering in the presence of an external confinement. Phys. Rev. Lett. 81, 938–941 (1998)

    ADS  CAS  Article  Google Scholar 

  11. Goerlitz, A. et al. Realization of Bose-Einstein condensates in lower dimensions. Phys. Rev. Lett. 87, 130402 (2001)

    ADS  Article  Google Scholar 

  12. Schreck, F. et al. A quasipure Bose-Einstein condensate immersed in a Fermi sea. Phys. Rev. Lett. 87, 080403 (2001)

    ADS  CAS  Article  Google Scholar 

  13. 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 (2001)

    ADS  CAS  Article  Google Scholar 

  14. Moritz, H., Stöferle, T., Köhl, M. & Esslinger, T. Exciting collective oscillations in a trapped 1D gas. Phys. Rev. Lett. 91, 250402 (2003)

    ADS  Article  Google Scholar 

  15. Laburthe Tolra, B., et al. Observation of reduced three-body recombination in a fermionized 1D Bose gas. Preprint at 〈〉 (2003)

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

    ADS  Article  Google Scholar 

  17. Efetov, K. B. & Larkin, A. I. Correlation functions in one-dimensional systems with strong interactions. Sov. Phys. JETP 42, 390–396 (1976)

    ADS  Google Scholar 

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

    Book  Google Scholar 

  19. Ovchinnikov, Y. B. et al. Diffraction of a released Bose-Einstein condensate by a pulsed standing light wave. Phys. Rev. Lett. 83, 284–287 (1999)

    ADS  CAS  Article  Google Scholar 

  20. Astrakharchik, G. E. & Giorgini, S. Correlation functions and momentum distributions of one-dimensional Bose systems. Phys. Rev. A 68, 031602 (2003)

    ADS  Article  Google Scholar 

  21. Olshanii, M. & Dunjko, V. Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases. Phys. Rev. Lett. 91, 090401 (2003)

    ADS  Article  Google Scholar 

  22. Cazalilla, M. A. Bosonizing one-dimensional cold atomic gases. J. Phys. B 37, S1–S47 (2004)

    ADS  CAS  Article  Google Scholar 

  23. 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  CAS  Article  Google Scholar 

  24. 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  CAS  Article  Google Scholar 

  25. 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  CAS  Article  Google Scholar 

  26. 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 

  27. Richard, S. et al. Momentum spectroscopy of 1D phase fluctuations in Bose-Einstein condensates. Phys. Rev. Lett. 91, 010405 (2003)

    ADS  CAS  Article  Google Scholar 

  28. Gangardt, D. M. & Shlyapnikov, G. V. Stability and phase coherence of trapped 1D Bose gases. Phys. Rev. Lett. 90, 010401 (2003)

    ADS  CAS  Article  Google Scholar 

  29. Paredes, B. & Cirac, J. I. From Cooper pairs to Luttinger liquids with bosonic atoms in optical lattices. Phys. Rev. Lett. 90, 150402 (2003)

    ADS  CAS  Article  Google Scholar 

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

    MATH  Google Scholar 

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We thank F. Gerbier, D. Gangardt and M. Olshanii for discussions, and M. Greiner for help in setting up the experiment. I.B. also acknowledges support from AFOSR.

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Correspondence to Immanuel Bloch.

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Supplementary information

Supplementary Figure 1

All twelve experimentally measured momentum profiles including comparison with our Fermionization based theory and comparison to calculations for ideal Bose and Fermi gases. (DOC 830 kb)

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

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