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Observation of first and second sound in a BKT superfluid


Superfluidity in its various forms has been of interest since the observation of frictionless flow in liquid helium II1,2. In three spatial dimensions it is conceptually associated with the emergence of long-range order at a critical temperature. One of the hallmarks of superfluidity, as predicted by the two-fluid model3,4 and observed in both liquid helium5 and in ultracold atomic gases6,7, is the existence of two kinds of sound excitation—the first and second sound. In two-dimensional systems, thermal fluctuations preclude long-range order8,9; however, superfluidity nevertheless emerges at a non-zero critical temperature through the infinite-order Berezinskii–Kosterlitz–Thouless (BKT) transition10,11, which is associated with a universal jump12 in the superfluid density without any discontinuities in the thermodynamic properties of the fluid. BKT superfluids are also predicted to support two sounds, but so far this has not been observed experimentally. Here we observe first and second sound in a homogeneous two-dimensional atomic Bose gas, and use the two temperature-dependent sound speeds to determine the superfluid density of the gas13,14,15,16. Our results agree with the predictions of BKT theory, including the prediction of a universal jump in the superfluid density at the critical temperature.

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Fig. 1: Sound excitations in a homogeneous two-dimensional Bose gas.
Fig. 2: First and second sound.
Fig. 3: The sound speeds and the superfluid density.

Data availability

The data that support the findings of this study are available in the Apollo repository ( Any additional information is available from the corresponding authors upon reasonable request. Source data are provided with this paper.


  1. 1.

    Kapitza, P. Viscosity of liquid helium below the λ-point. Nature 141, 74 (1938).

    ADS  CAS  Google Scholar 

  2. 2.

    Allen, J. F. & Misener, A. D. Flow of liquid helium II. Nature 141, 75 (1938).

    ADS  CAS  Google Scholar 

  3. 3.

    Tisza, L. Transport phenomena in helium II. Nature 141, 913 (1938).

    ADS  CAS  Google Scholar 

  4. 4.

    Landau, L. Theory of the superfluidity of helium II. Phys. Rev. 60, 356–358 (1941).

    ADS  CAS  MATH  Google Scholar 

  5. 5.

    Peshkov, V. Second sound in helium II. Sov. Phys. JETP 11, 580–584 (1960).

    Google Scholar 

  6. 6.

    Stamper-Kurn, D. M., Miesner, H.-J., Inouye, S. & Andrews, M. R. & Ketterle, W. Collisionless and hydrodynamic excitations of a Bose–Einstein condensate. Phys. Rev. Lett. 81, 500–503 (1998).

    ADS  CAS  Google Scholar 

  7. 7.

    Sidorenkov, L. A. et al. Second sound and the superfluid fraction in a Fermi gas with resonant interactions. Nature 498, 78–81 (2013).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  8. 8.

    Hohenberg, P. C. Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383–386 (1967).

    ADS  CAS  Google Scholar 

  9. 9.

    Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1307 (1966).

    ADS  Google Scholar 

  10. 10.

    Berezinskii, V. L. Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continous symmetry group. II. Quantum systems. Sov. Phys. JETP 34, 610 (1971).

    ADS  MathSciNet  Google Scholar 

  11. 11.

    Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in twodimensional systems. J. Phys. C 6, 1181–1203 (1973).

    ADS  CAS  Google Scholar 

  12. 12.

    Nelson, D. R. & Kosterlitz, J. M. Universal jump in the superfluid density of two-dimensional superfluids. Phys. Rev. Lett. 39, 1201–1205 (1977).

    ADS  CAS  Google Scholar 

  13. 13.

    Prokof’ev, N., Ruebenacker, O. & Svistunov, B. Critical point of a weakly interacting two-dimensional Bose gas. Phys. Rev. Lett. 87, 270402 (2001).

    PubMed  PubMed Central  Google Scholar 

  14. 14.

    Prokof’ev, N. & Svistunov, B. Two-dimensional weakly interacting Bose gas in the fluctuation region. Phys. Rev. A 66, 043608 (2002).

    ADS  Google Scholar 

  15. 15.

    Ozawa, T. & Stringari, S. Discontinuities in the first and second sound velocities at the Berezinskii–Kosterlitz–Thouless transition. Phys. Rev. Lett. 112, 025302 (2014).

    ADS  Google Scholar 

  16. 16.

    Ota, M. & Stringari, S. Second sound in a two-dimensional Bose gas: from the weakly to the strongly interacting regime. Phys. Rev. A 97, 033604 (2018).

    ADS  CAS  Google Scholar 

  17. 17.

    Hu, H., Taylor, E., Liu, X.-J., Stringari, S. & Griffin, A. Second sound and the density response function in uniform superfluid atomic gases. New J. Phys. 12, 043040 (2010).

    ADS  Google Scholar 

  18. 18.

    Bishop, D. J. & Reppy, J. D. Study of the superfluid transition in two-dimensional 4He films. Phys. Rev. Lett. 40, 1727–1730 (1978).

    ADS  CAS  Google Scholar 

  19. 19.

    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  CAS  PubMed  PubMed Central  Google Scholar 

  20. 20.

    Cladé, P., Ryu, C., Ramanathan, A., Helmerson, K. & Phillips, W. D. Observation of a 2D Bose gas: from thermal to quasicondensate to superfluid. Phys. Rev. Lett. 102, 170401 (2009).

    ADS  PubMed  PubMed Central  Google Scholar 

  21. 21.

    Tung, S., Lamporesi, G., Lobser, D., Xia, L. & Cornell, E. A. Observation of the presuperfluid regime in a two-dimensional Bose gas. Phys. Rev. Lett. 105, 230408 (2010).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  22. 22.

    Yefsah, T., Desbuquois, R., Chomaz, L., Günter, K. J. & Dalibard, J. Exploring the thermodynamics of a two-dimensional Bose gas. Phys. Rev. Lett. 107, 130401 (2011).

    ADS  PubMed  PubMed Central  Google Scholar 

  23. 23.

    Hung, C.-L., Zhang, X., Gemelke, N. & Chin, C. Observation of scale invariance and universality in two-dimensional Bose gases. Nature 470, 236–239 (2011).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  24. 24.

    Hadzibabic, Z. & Dalibard, J. Two-dimensional Bose fluids: an atomic physics perspective. Riv. Nuovo Cimento 34, 389–434 (2011).

    CAS  Google Scholar 

  25. 25.

    Desbuquois, R. et al. Superfluid behaviour of a two-dimensional Bose gas. Nat. Phys. 8, 645–648 (2012).

    CAS  Google Scholar 

  26. 26.

    Ha, L.-C. et al. Strongly interacting two-dimensional Bose gases. Phys. Rev. Lett. 110, 145302 (2013).

    ADS  PubMed  PubMed Central  Google Scholar 

  27. 27.

    Choi, J. Y., Seo, S. W. & Shin, Y. I. Observation of thermally activated vortex pairs in a quasi-2D Bose gas. Phys. Rev. Lett. 110, 175302 (2013).

    ADS  PubMed  PubMed Central  Google Scholar 

  28. 28.

    Chomaz, L. et al. Emergence of coherence via transverse condensation in a uniform quasi-two-dimensional Bose gas. Nat. Commun. 6, 6162 (2015).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  29. 29.

    Fletcher, R. J. et al. Connecting Berezinskii–Kosterlitz–Thouless and BEC phase transitions by tuning interactions in a trapped gas. Phys. Rev. Lett. 114, 255302 (2015).

    ADS  MathSciNet  PubMed  PubMed Central  Google Scholar 

  30. 30.

    Murthy, P. A. et al. Observation of the Berezinskii–Kosterlitz–Thouless phase transition in an ultracold Fermi gas. Phys. Rev. Lett. 115, 010401 (2015).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  31. 31.

    Ville, J. L. et al. Sound propagation in a uniform superfluid two-dimensional Bose gas. Phys. Rev. Lett. 121, 145301 (2018).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  32. 32.

    Ota, M. et al. Collisionless sound in a uniform two-dimensional Bose gas. Phys. Rev. Lett. 121, 145302 (2018).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  33. 33.

    Cappellaro, A., Toigo, F. & Salasnich, L. Collisionless dynamics in two-dimensional bosonic gases. Phys. Rev. A 98, 043605 (2018).

    ADS  CAS  Google Scholar 

  34. 34.

    Wu, Z., Zhang, S. & Zhai, H. Dynamic Kosterlitz–Thouless theory for two-dimensional ultracold atomic gases. Phys. Rev. A 102, 043311 (2020).

    ADS  CAS  Google Scholar 

  35. 35.

    Bohlen, M. et al. Sound propagation and quantum-limited damping in a two-dimensional Fermi gas. Phys. Rev. Lett. 124, 240403 (2020).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  36. 36.

    Petrov, D. S., Holzmann, M. & Shlyapnikov, G. V. Bose–Einstein condensation in quasi-2D trapped gases. Phys. Rev. Lett. 84, 2551–2555 (2000).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  37. 37.

    Fletcher, R. J. et al. Elliptic flow in a strongly interacting normal Bose gas. Phys. Rev. A 98, 011601 (2018).

    ADS  CAS  Google Scholar 

  38. 38.

    Navon, N., Gaunt, A. L., Smith, R. P. & Hadzibabic, Z. Emergence of a turbulent cascade in a quantum gas. Nature 539, 72–75 (2016).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  39. 39.

    Pitaevskii, L. & Stringari, S. Bose–Einstein Condensation and Superfluidity Ch. 7 (Oxford Univ. Press, 2016).

  40. 40.

    Hohenberg, P. C. & Martin, P. C. Superfluid dynamics in the hydrodynamic (ωτ 1) and collisionless (ωτ 1) domains. Phys. Rev. Lett. 12, 69–71(1964).

    ADS  Google Scholar 

  41. 41.

    Singh, V. P. & Mathey, L. Sound propagation in a two-dimensional Bose gas across the superfluid transition. Phys. Rev. Res. 2, 023336 (2020).

    CAS  Google Scholar 

  42. 42.

    Patel, P. B. et al. Universal sound diffusion in a strongly interacting Fermi gas. Science 370, 1222–1226 (2020).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  43. 43.

    Pilati, S., Giorgini, S. & Prokof’ev, N. Critical temperature of interacting Bose gases in two and three dimensions. Phys. Rev. Lett. 100, 140405 (2008).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  44. 44.

    Foster, C. J., Blakie, P. B. & Davis, M. J. Vortex pairing in two-dimensional Bose gases. Phys. Rev. A 81, 023623 (2010).

    ADS  Google Scholar 

  45. 45.

    Gawryluk, K. & Brewczyk, M. Signatures of a universal jump in the superfluid density of a two-dimensional Bose gas with a finite number of particles. Phys. Rev. A 99, 033615 (2019).

    ADS  CAS  Google Scholar 

  46. 46.

    Lee, T. D. & Yang, C. N. Low-temperature behavior of a dilute Bose system of hard spheres. II. Nonequilibrium properties. Phys. Rev. 113, 1406–1413 (1959).

    ADS  MathSciNet  MATH  Google Scholar 

  47. 47.

    Pitaevskii, L. & Stringari, S. In Universal Themes of Bose-Einstein Condensation (eds Proukakis, N. P. et al.) 322–347 (Cambridge Univ. Press, 2017).

  48. 48.

    Eigen, C. et al. Observation of weak collapse in a Bose–Einstein condensate. Phys. Rev. X 6, 041058 (2016).

    Google Scholar 

  49. 49.

    Campbell, R. L. D. et al. Efficient production of large 39K Bose–Einstein condensates. Phys. Rev. A 82, 063611 (2010).

    ADS  Google Scholar 

  50. 50.

    Zaccanti, M. et al. Observation of an Efimov spectrum in an atomic system. Nat. Phys. 5, 586 (2009).

    CAS  Google Scholar 

  51. 51.

    Hohenberg, P. C. & Martin, P. C. Microscopic theory of superfluid helium. Ann. Phys. 34, 291 (1965).

    ADS  CAS  Google Scholar 

  52. 52.

    Hu, H., Zou, P. & Liu, X.-J. Low-momentum dynamic structure factor of a strongly interacting Fermi gas at finite temperature: a two-fluid hydrodynamic description. Phys. Rev. A 97, 023615 (2018).

    ADS  CAS  Google Scholar 

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We thank J. Man for experimental assistance; and R. P. Smith, J. Dalibard, M. Zwierlein, R. J. Fletcher, T. A. Hilker, S. Nascimbene and T. Yefsah for discussions. This work was supported by the EPSRC (grant nos EP/N011759/1 and EP/P009565/1), ERC (QBox) and QuantERA (NAQUAS, EPSRC grant no. EP/R043396/1). J.S. acknowledges support from Churchill College (Cambridge), and Z.H. acknowledges support from the Royal Society Wolfson Fellowship.

Author information




P.C. led the data acquisition and analysis. M.G. and J.S. contributed to the data acquisition. P.C., M.G., R.L. and J.S. contributed to the experimental setup. P.C., N.D. and J.S. produced the figures. Z.H. supervised the project. All authors contributed to the data analysis, interpretation of the results and writing of the manuscript.

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Correspondence to Panagiotis Christodoulou.

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The authors declare no competing interests.

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Christodoulou, P., Gałka, M., Dogra, N. et al. Observation of first and second sound in a BKT superfluid. Nature 594, 191–194 (2021).

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