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# Observation of first and second sound in a BKT superfluid

## Abstract

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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## Data availability

The data that support the findings of this study are available in the Apollo repository (https://doi.org/10.17863/CAM.66056). Any additional information is available from the corresponding authors upon reasonable request. Source data are provided with this paper.

## References

1. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

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).

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).

8. 8.

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

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).

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).

11. 11.

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

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).

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).

14. 14.

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

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).

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).

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).

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).

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).

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).

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).

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).

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).

24. 24.

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

25. 25.

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

26. 26.

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

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).

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).

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).

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).

31. 31.

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

32. 32.

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

33. 33.

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

34. 34.

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

35. 35.

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

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).

37. 37.

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

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).

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).

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).

42. 42.

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

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).

44. 44.

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

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).

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).

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).

49. 49.

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

50. 50.

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

51. 51.

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

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).

## Acknowledgements

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

Authors

### Contributions

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.

### Corresponding author

Correspondence to Panagiotis Christodoulou.

## Ethics declarations

### Competing interests

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). https://doi.org/10.1038/s41586-021-03537-9

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