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Thermodynamics of a deeply degenerate SU(N)-symmetric Fermi gas


Many-body quantum systems can exhibit a striking degree of symmetry unparallelled in their classical counterparts. In real materials SU(N) symmetry is an idealization, but this symmetry is pristinely realized in fully controllable ultracold alkaline-earth atomic gases. Here, we study an SU(N)-symmetric Fermi liquid of 87Sr atoms, where N can be tuned to be as large as 10. In the deeply degenerate regime, we show through precise measurements of density fluctuations and expansion dynamics that the large N of spin states under SU(N) symmetry leads to pronounced interaction effects in a system with a nominally negligible interaction parameter. Accounting for these effects, we demonstrate thermometry accurate to 1% of the Fermi energy. We also demonstrate record speed for preparing degenerate Fermi seas enabled by the SU(N)-symmetric interactions, reaching T/TF = 0.22 with 10 nuclear spin states in 0.6 s working with a laser-cooled sample. This, along with the introduction of a new spin polarizing method, enables the operation of a three-dimensional optical lattice clock in the band insulating regime.

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Fig. 1: Degenerate bosons, fermions and SU(N) fermions.
Fig. 2: Preparation of a degenerate Fermi gas.
Fig. 3: Local density fluctuations of an SU(N) degenerate gas.
Fig. 4: Cloud anisotropy.
Fig. 5: Interaction signatures.

Data availability

The datasets generated and analysed during the current study are available from the corresponding author L.S. on reasonable request.


  1. 1.

    Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).

    ADS  Google Scholar 

  2. 2.

    Inguscio, M., Ketterle, W. & Salomon, C. (eds) Making, Probing and Understanding Ultracold Fermi Gases (IOS, 2008).

  3. 3.

    Cazalilla, M. & Rey, A. M. Ultracold fermi gases with emergent SU(N) symmetry. Rep. Prog. Phys. 77, 124401 (2014).

    ADS  Google Scholar 

  4. 4.

    Horikoshi, M., Nakajima, S., Ueda, M. & Mukaiyama, T. Measurement of universal thermodynamic functions for a unitary Fermi gas. Science 327, 442–445 (2010).

    ADS  Google Scholar 

  5. 5.

    Nascimbéne, S., Navon, N., Jiang, K., Chevy, F. & Salomon, C. Exploring the thermodynamics of a universal Fermi gas. Nature 463, 1057–1060 (2010).

    ADS  Google Scholar 

  6. 6.

    Gorshkov, A. V. et al. Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms. Nat. Phys. 6, 289–295 (2010).

    Google Scholar 

  7. 7.

    Taie, S., Yamazaki, R., Sugawa, S. & Takahashi, Y. An SU(6) Mott insulator of an atomic Fermi gas realized by large-spin Pomeranchuk cooling. Nat. Phys. 8, 825–830 (2012).

    Google Scholar 

  8. 8.

    Scazza, F. et al. Observation of two-orbital spin-exchange interactions with ultracold SU(N)-symmetric fermions. Nat. Phys. 10, 779–784 (2014).

    Google Scholar 

  9. 9.

    Hofrichter, C. et al. Direct probing of the Mott crossover in the SU(N) Fermi-Hubbard model. Phys. Rev. X 6, 021030 (2016).

    Google Scholar 

  10. 10.

    Goban, A. et al. Emergence of multi-body interactions in a fermionic lattice clock. Nature 563, 369–373 (2018).

    ADS  Google Scholar 

  11. 11.

    Ozawa, H., Taie, S. & Takahashi, Y. Antiferromagnetic spin correlation of SU(N) Fermi gas in an optical lattice. Phys. Rev. Lett. 121, 225303 (2018).

    ADS  Google Scholar 

  12. 12.

    Zhang, X. et al. Spectroscopic observation of SU(N)-symmetric interactions in Sr orbital magnetism. Science 345, 1467–1473 (2014).

    ADS  Google Scholar 

  13. 13.

    Pagano, G. et al. A one-dimensional liquid of fermions with tunable spin. Nat. Phys. 10, 198–201 (2014).

    Google Scholar 

  14. 14.

    Song, B. et al. Evidence for bosonization in a three-dimensional gas of SU(N) fermions. Preprint at (2020).

  15. 15.

    He, C. et al. Collective excitations in two-dimensional SU(N) Fermi gases with tunable spin. Phys. Rev. Res. 2, 012028 (2020).

    Google Scholar 

  16. 16.

    Stellmer, S., Schreck, F. & Killian, T. C. in Annual Review of Cold Atoms and Molecules Ch. 1 (World Scientific, 2014).

  17. 17.

    Hazzard, K. R. A., Gurarie, V., Hermele, M. & Rey, A. M. High-temperature properties of fermionic alkaline-earth-metal atoms in optical lattices. Phys. Rev. A 85, 041604 (2012).

    ADS  Google Scholar 

  18. 18.

    Bonnes, L., Hazzard, K. R. A., Manmana, S. R., Rey, A. M. & Wessel, S. Adiabatic loading of one-dimensional SU(N) alkaline-earth-atom fermions in optical lattices. Phys. Rev. Lett. 109, 205305 (2012).

    ADS  Google Scholar 

  19. 19.

    Messio, L. & Mila, F. Entropy dependence of correlations in one-dimensional SU(N) antiferromagnets. Phys. Rev. Lett. 109, 205306 (2012).

    ADS  Google Scholar 

  20. 20.

    Yip, S.-K., Huang, J. & Kao, J. Theory of SU(N) fermi liquids. Phys. Rev. A 89, 043610 (2014).

    ADS  Google Scholar 

  21. 21.

    Campbell, S. L. et al. A Fermi-degenerate three-dimensional optical lattice clock. Science 358, 90–94 (2017).

    ADS  Google Scholar 

  22. 22.

    Marti, G. E. et al. Imaging optical frequencies with 100μHz precision and 1.1μm resolution. Phys. Rev. Lett. 120, 103201 (2018).

    ADS  Google Scholar 

  23. 23.

    Oelker, E. et al. Demonstration of 4.8 × 1017 stability at 1 s for two independent optical clocks. Nat. Photon. 13, 714–719 (2019).

    ADS  Google Scholar 

  24. 24.

    McGrew, W. F. et al. Atomic clock performance enabling geodesy below the centimetre level. Nature 564, 87–90 (2018).

    ADS  Google Scholar 

  25. 25.

    Huntemann, N., Sanner, C., Lipphardt, B., Tamm, C. & Peik, E. Single-ion atomic clock with 3 × 10−18 systematic uncertainty. Phys. Rev. Lett. 116, 063001 (2016).

  26. 26.

    Sanner, C. et al. Optical clock comparison for Lorentz symmetry testing. Nature 567, 204–208 (2019).

    ADS  Google Scholar 

  27. 27.

    Kolkowitz, S. et al. Gravitational wave detection with optical lattice atomic clocks. Phys. Rev. D 94, 124043 (2016).

    ADS  Google Scholar 

  28. 28.

    Xu, V., Jaffe, M., Panda, C., Clark, L. & Müller, H. Probing gravity by holding atoms for 20 seconds. Science 366, 745–749 (2019).

  29. 29.

    Schioppo, M. et al. Ultrastable optical clock with two cold-atom ensembles. Nat. Photon. 11, 48–52 (2017).

    ADS  Google Scholar 

  30. 30.

    Hutson, R. B. et al. Engineering quantum states of matter for atomic clocks in shallow optical lattices. Phys. Rev. Lett. 123, 123401 (2019).

    ADS  Google Scholar 

  31. 31.

    Loftus, T., Ido, T., Ludlow, A., Boyd, M. & Ye, J. Narrow line cooling: finite photon recoil dynamics. Phys. Rev. Lett. 93, 073003 (2004).

    ADS  Google Scholar 

  32. 32.

    Stellmer, S., Grimm, R. & Schreck, F. Production of quantum-degenerate strontium gases. Phys. Rev. A 87, 013611 (2013).

    ADS  Google Scholar 

  33. 33.

    Julienne, P. S., Smith, A. M. & Burnett, K. in Advances In Atomic, Molecular, and Optical Physics Vol. 30 (eds Bates, D. & Bederson, B.) 141–198 (Academic, 1992).

  34. 34.

    Sesko, D. W., Walker, T. G. & Wieman, C. E. Behavior of neutral atoms in a spontaneous force trap. J. Opt. Soc. Am. B 8, 946–958 (1991).

    ADS  Google Scholar 

  35. 35.

    Stellmer, S., Pasquiou, B., Grimm, R. & Schreck, F. Laser cooling to quantum degeneracy. Phys. Rev. Lett. 110, 263003 (2013).

    ADS  Google Scholar 

  36. 36.

    Mukaiyama, T., Katori, H., Ido, T., Li, Y. & Kuwata-Gonokami, M. Recoil-limited laser cooling of 87Sr atoms near the Fermi temperature. Phys. Rev. Lett. 90, 113002 (2003).

    ADS  Google Scholar 

  37. 37.

    Fukuhara, T., Takasu, Y., Kumakura, M. & Takahashi, Y. Degenerate Fermi gases of ytterbium. Phys. Rev. Lett. 98, 030401 (2007).

    ADS  Google Scholar 

  38. 38.

    Dick, G. J. Local Oscillator Induced Instabilities in Trapped Ion Frequency Standards. In Proceedings of the 19th Annual Precise Time and Time Interval Meeting, 133-147 (US Naval Observatory, 1988).

  39. 39.

    Sleator, T., Pfau, T., Balykin, V., Carnal, O. & Mlynek, J. Experimental demonstration of the optical Stern-Gerlach effect. Phys. Rev. Lett. 68, 1996–1999 (1992).

    ADS  Google Scholar 

  40. 40.

    Taie, S. et al. Realization of a SU(2) × SU(6) system of fermions in a cold atomic gas. Phys. Rev. Lett. 105, 190401 (2010).

  41. 41.

    Stellmer, S., Grimm, R. & Schreck, F. Detection and manipulation of nuclear spin states in fermionic strontium. Phys. Rev. A 84, 043611 (2011).

    ADS  Google Scholar 

  42. 42.

    Lee, Y.-R. et al. Compressibility of an ultracold Fermi gas with repulsive interactions. Phys. Rev. A 85, 063615 (2012).

    ADS  Google Scholar 

  43. 43.

    Callen, H. & T.A., W. Irreversibility and generalized noise. Phys. Rev. 83, 34–40 (1951).

    ADS  MathSciNet  MATH  Google Scholar 

  44. 44.

    Sanner, C. et al. Suppression of density fluctuations in a quantum degenerate Fermi gas. Phys. Rev. Lett. 105, 040402 (2010).

    ADS  Google Scholar 

  45. 45.

    Müller, T. et al. Local observation of antibunching in a trapped Fermi gas. Phys. Rev. Lett. 105, 040401 (2010).

    ADS  Google Scholar 

  46. 46.

    Danielewicz, P. Quantum theory of nonequilibrium processes, I. Ann. Phys. 152, 239–304 (1984).

    ADS  Google Scholar 

  47. 47.

    Kadanoff, L. P., Baym, G. & Pines, D. Quantum Statistical Mechanics 1st edn (CRC, 2019).

  48. 48.

    Pedri, P., Guéry-Odelin, D. & Stringari, S. Dynamics of a classical gas including dissipative and mean-field effects. Phys. Rev. A 68, 043608 (2003).

    ADS  Google Scholar 

  49. 49.

    Giorgini, S., Pitaevskii, L. P. & Stringari, S. Theory of ultracold atomic Fermi gases. Rev. Mod. Phys. 80, 1215–1274 (2008).

    ADS  Google Scholar 

  50. 50.

    Menotti, C., Pedri, P. & Stringari, S. Expansion of an interacting fermi gas. Phys. Rev. Lett. 89, 250402 (2002).

    ADS  Google Scholar 

  51. 51.

    Castin, Y. & Dum, R. Bose-Einstein condensates in time dependent traps. Phys. Rev. Lett. 77, 5315–5319 (1996).

    ADS  Google Scholar 

  52. 52.

    Guéry-Odelin, D. Mean-field effects in a trapped gas. Phys. Rev. A 66, 033613 (2002).

    ADS  Google Scholar 

  53. 53.

    O’Hara, K. M., Hemmer, S. L., Gehm, M. E., Granade, S. R. & Thomas, J. E. Observation of a strongly interacting degenerate Fermi gas of atoms. Science 298, 2179–2182 (2002).

    ADS  Google Scholar 

  54. 54.

    Burt, E. A. et al. Coherence, correlations, and collisions: what one learns about Bose-Einstein condensates from their decay. Phys. Rev. Lett. 79, 337–340 (1997).

    ADS  Google Scholar 

  55. 55.

    Wolf, J. et al. State-to-state chemistry for three-body recombination in an ultracold rubidium gas. Science 358, 921–924 (2017).

    ADS  Google Scholar 

  56. 56.

    Vichi, L. Collisional Damping of the Collective Oscillations of a Trapped Fermi Gas. J. Low Temp. Phys. 121, 177–197 (2000).

    ADS  Google Scholar 

  57. 57.

    Jackson, B., Pedri, P. & Stringari, S. Collisions and expansion of an ultracold dilute Fermi gas. Europhys. Lett. 67, 524–530 (2004).

    ADS  Google Scholar 

  58. 58.

    Lee, T. D. & Yang, C. N. Many-body problem in quantum statistical mechanics. I. General formulation. Phys. Rev. 113, 1165–1177 (1959).

    ADS  MathSciNet  MATH  Google Scholar 

  59. 59.

    Lee, T. D. & Yang, C. N. Many-body problem in quantum statistical mechanics. II. Virial expansion for hard-sphere gas. Phys. Rev. 116, 25–31 (1959).

    ADS  MathSciNet  MATH  Google Scholar 

  60. 60.

    Lee, T. D. & Yang, C. N. Many-body problem in quantum statistical mechanics. III. Zero-temperature limit for dilute hard spheres. Phys. Rev. 117, 12–21 (1960).

    ADS  MathSciNet  MATH  Google Scholar 

  61. 61.

    Pathria, R. K. & Kawatra, M. P. Quantum statistical mechanics of a many-body system with square-well interaction. Progr. Theor. Phys. 27, 638–652 (1962).

    ADS  MathSciNet  MATH  Google Scholar 

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We thank T. Bothwell, C. Kennedy, E. Oelker and J. Robinson for discussions and technical contributions. We also thank J. Thompson and C. Kennedy for reading the manuscript. This work is supported by NIST, DARPA, and AFOSR grant nos. FA9550-19-1-0275 and FA9550-18-1-0319, and NSF grant no. PHYS-1734006. C.S. thanks the Humboldt Foundation for support.

Author information




L.S., C.S., R.B.H., A.G., L.Y., W.R.M. and J.Y. contributed to the experimental measurements. T.B. and A.M.R. developed the theoretical model. All authors discussed the results, contributed to the data analysis and worked together on the manuscript.

Corresponding authors

Correspondence to Lindsay Sonderhouse or Jun Ye.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Stefano Giorgini, Florian Schreck and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Sonderhouse, L., Sanner, C., Hutson, R.B. et al. Thermodynamics of a deeply degenerate SU(N)-symmetric Fermi gas. Nat. Phys. 16, 1216–1221 (2020).

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