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.
Subscribe to Journal
Get full journal access for 1 year
only $3.83 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Girardeau, M. Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. Phys. 1, 516–523 (1960)
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)
Lenard, A. Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons. J. Math. Phys. 5, 930–943 (1964)
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)
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)
Jochim, S. et al. Bose-Einstein condensation of molecules. Science 302, 2101–2103 (2003)
Greiner, M., Regal, C. & Jin, D. S. Emergence of a molecular Bose-Einstein condensate from a Fermi gas. Nature 426, 537–540 (2003)
Zwierlein, M. W. et al. Observation of Bose-Einstein condensation of molecules. Phys. Rev. Lett. 91, 250401 (2003)
Regal, C., Greiner, M. & Jin, D. S. Observation of resonance condensation of fermionic atom pairs. Phys. Rev. Lett. 92, 040403 (2004)
Olshanii, M. Atomic scattering in the presence of an external confinement. Phys. Rev. Lett. 81, 938–941 (1998)
Goerlitz, A. et al. Realization of Bose-Einstein condensates in lower dimensions. Phys. Rev. Lett. 87, 130402 (2001)
Schreck, F. et al. A quasipure Bose-Einstein condensate immersed in a Fermi sea. Phys. Rev. Lett. 87, 080403 (2001)
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)
Moritz, H., Stöferle, T., Köhl, M. & Esslinger, T. Exciting collective oscillations in a trapped 1D gas. Phys. Rev. Lett. 91, 250402 (2003)
Laburthe Tolra, B., et al. Observation of reduced three-body recombination in a fermionized 1D Bose gas. Preprint at 〈http://xxx.lanl.gov/cond-mat/0312003〉 (2003)
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)
Efetov, K. B. & Larkin, A. I. Correlation functions in one-dimensional systems with strong interactions. Sov. Phys. JETP 42, 390–396 (1976)
Korepin, V. E., Bogoliubov, N. M. & Izergin, A. G. Quantum Inverse Scattering Method and Correlation Functions (Cambridge Univ. Press, Cambridge, 1993)
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)
Astrakharchik, G. E. & Giorgini, S. Correlation functions and momentum distributions of one-dimensional Bose systems. Phys. Rev. A 68, 031602 (2003)
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)
Cazalilla, M. A. Bosonizing one-dimensional cold atomic gases. J. Phys. B 37, S1–S47 (2004)
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)
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)
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)
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)
Richard, S. et al. Momentum spectroscopy of 1D phase fluctuations in Bose-Einstein condensates. Phys. Rev. Lett. 91, 010405 (2003)
Gangardt, D. M. & Shlyapnikov, G. V. Stability and phase coherence of trapped 1D Bose gases. Phys. Rev. Lett. 90, 010401 (2003)
Paredes, B. & Cirac, J. I. From Cooper pairs to Luttinger liquids with bosonic atoms in optical lattices. Phys. Rev. Lett. 90, 150402 (2003)
Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, Cambridge, 1999)
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.
The authors declare that they have no competing financial interests.
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)
About this article
Cite this article
Paredes, B., Widera, A., Murg, V. et al. Tonks–Girardeau gas of ultracold atoms in an optical lattice. Nature 429, 277–281 (2004). https://doi.org/10.1038/nature02530
Few-Body Systems (2020)
Journal of Low Temperature Physics (2019)
New Journal of Physics (2019)
Physical Review Letters (2019)
New Journal of Physics (2019)