Evidence for universal relations describing a gas with p-wave interactions


In dilute gases, a set of universal relations, known as the contact relations, directly connects thermodynamics and microscopic properties. So far, they have been established only for interactions with s-wave symmetry—that is, without relative angular momentum. Here we report measurements of two new physical quantities, the p-wave contacts, and, using recently proposed relations, present evidence that they encode the universal aspects of p-wave interactions. Our experiments use an ultracold Fermi gas of 40K, in which s-wave interactions are suppressed by polarizing the sample, whereas p-wave interactions are enhanced by working near a scattering resonance. Using time-resolved spectroscopy, we study how correlations in the system develop after quenching the atoms into an interacting state. By combining quasi-steady-state measurements with new contact relations, we infer an attractive p-wave interaction energy as large as half the Fermi energy. Our results reveal new ways to understand and characterize the properties of a resonant p-wave quantum gas.

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Figure 1: Tuning p-wave interactions in 40K.
Figure 2: Observation of the p-wave contacts.
Figure 3: The p-wave contacts near two Feshbach resonances.
Figure 4: Dynamics of the p-wave contact and atom number close to resonance.


  1. 1

    Tan, S. Energetics of a strongly correlated Fermi gas. Ann. Phys. 323, 2952–2970 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  2. 2

    Tan, S. Large momentum part of a strongly correlated Fermi gas. Ann. Phys. 323, 2971–2986 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  3. 3

    Tan, S. Generalized virial theorem and pressure relation for a strongly correlated Fermi gas. Ann. Phys. 323, 2987–2990 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  4. 4

    Werner, F., Tarruell, L. & Castin, Y. Number of closed-channel molecules in the BEC–BCS crossover. Eur. Phys. J. B 68, 401–415 (2009).

    ADS  Google Scholar 

  5. 5

    Zhang, S. & Leggett, A. J. Universal properties of the ultracold Fermi gas. Phys. Rev. A 79, 023601 (2009).

    ADS  Google Scholar 

  6. 6

    Braaten, E. in The BCS–BEC Crossover and the Unitary Fermi Gas (ed. Zwerger, W.) 193–231 (Springer, 2012).

    Google Scholar 

  7. 7

    Werner, F. & Castin, Y. General relations for quantum gases in two and three dimensions: two-component fermions. Phys. Rev. A 86, 013626 (2012).

    ADS  Google Scholar 

  8. 8

    Werner, F. & Castin, Y. General relations for quantum gases in two and three dimensions. II. Bosons and mixtures. Phys. Rev. A 86, 053633 (2012).

    ADS  Google Scholar 

  9. 9

    Wild, R. J., Makotyn, P., Pino, J. M., Cornell, E. A. & Jin, D. S. Measurements of Tan’s contact in an atomic Bose–Einstein condensate. Phys. Rev. Lett. 108, 145305 (2012).

    ADS  Google Scholar 

  10. 10

    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  Google Scholar 

  11. 11

    Combescot, R., Alzetto, F. & Leyronas, X. Particle distribution tail and related energy formula. Phys. Rev. A 79, 053640 (2009).

    ADS  Google Scholar 

  12. 12

    Fröhlich, B. et al. Two-dimensional Fermi liquid with attractive interactions. Phys. Rev. Lett. 109, 130403 (2012).

    ADS  Google Scholar 

  13. 13

    Barth, M. & Zwerger, W. Tan relations in one dimension. Ann. Phys. 326, 2544–2565 (2011).

    ADS  MATH  Google Scholar 

  14. 14

    Weiss, R., Bazak, B. & Barnea, N. Nuclear neutron–proton contact and the photoabsorption cross section. Phys. Rev. Lett. 114, 012501 (2015).

    ADS  Google Scholar 

  15. 15

    DeMarco, B., Bohn, J. L., Burke, J. P., Holland, M. & Jin, D. S. Measurement of p-wave threshold law using evaporatively cooled fermionic atoms. Phys. Rev. Lett. 82, 4208–4211 (1999).

    ADS  Google Scholar 

  16. 16

    Regal, C. A., Ticknor, C., Bohn, J. L. & Jin, D. S. Tuning p-wave interactions in an ultracold Fermi gas of atoms. Phys. Rev. Lett. 90, 053201 (2003).

    ADS  Google Scholar 

  17. 17

    Zhang, J. et al. p-wave Feshbach resonances of ultracold 6Li. Phys. Rev. A 70, 030702 (2004).

    ADS  Google Scholar 

  18. 18

    Schunck, C. H. et al. Feshbach resonances in fermionic Li-6. Phys. Rev. A 71, 045601 (2005).

    ADS  Google Scholar 

  19. 19

    Kallin, C. Chiral p-wave order in Sr2RuO4 . Rep. Prog. Phys. 75, 042501 (2012).

    ADS  Google Scholar 

  20. 20

    Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000).

    ADS  Google Scholar 

  21. 21

    Levinsen, J., Cooper, N. R. & Gurarie, V. Strongly resonant p-wave superfluids. Phys. Rev. Lett. 99, 210402 (2007).

    ADS  Google Scholar 

  22. 22

    Elliott, S. R. & Franz, M. Majorana fermions in nuclear, particle, and solid-state physics. Rev. Mod. Phys. 87, 137–163 (2015).

    ADS  MathSciNet  Google Scholar 

  23. 23

    Inotani, D., Watanabe, R., Sigrist, M. & Ohashi, Y. Pseudogap phenomenon in an ultracold Fermi gas with a p-wave pairing interaction. Phys. Rev. A 85, 053628 (2012).

    ADS  Google Scholar 

  24. 24

    Yoshida, S. M. & Ueda, M. Universal high-momentum asymptote and thermodynamic relations in a spinless Fermi gas with a resonant p-wave interaction. Phys. Rev. Lett. 115, 135303 (2015).

    ADS  Google Scholar 

  25. 25

    Yu, Z., Thywissen, J. H. & Zhang, S. Universal relations for a Fermi gas close to a p-wave interaction resonance. Phys. Rev. Lett. 115, 135304 (2015).

    ADS  Google Scholar 

  26. 26

    He, M.-Y., Zhang, S.-L., Chan, H. M. & Zhou, Q. Concept of contact spectrum and its applications in atomic quantum Hall states. Phys. Rev. Lett. 116, 045301 (2016).

    ADS  Google Scholar 

  27. 27

    Zhang, P., Naidon, P. & Ueda, M. Scattering amplitude of ultracold atoms near the p-wave magnetic Feshbach resonance. Phys. Rev. A 82, 062712 (2010).

    ADS  Google Scholar 

  28. 28

    Ticknor, C., Regal, C. A., Jin, D. S. & Bohn, J. L. Multiplet structure of Feshbach resonances in nonzero partial waves. Phys. Rev. A 69, 042712 (2004).

    ADS  Google Scholar 

  29. 29

    Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225–1286 (2010).

    ADS  Google Scholar 

  30. 30

    Jona-Lasinio, M., Pricoupenko, L. & Castin, Y. Three fully polarized fermions close to a p-wave Feshbach resonance. Phys. Rev. A 77, 043611 (2008).

    ADS  Google Scholar 

  31. 31

    Günter, K., Stöferle, T., Moritz, H., Köhl, M. & Esslinger, T. p-wave interactions in low-dimensional fermionic gases. Phys. Rev. Lett. 95, 230401 (2005).

    ADS  Google Scholar 

  32. 32

    Peng, S.-G., Tan, S. & Jiang, K. Manipulation of p-wave scattering of cold atoms in low dimensions using the magnetic field vector. Phys. Rev. Lett. 112, 250401 (2014).

    ADS  Google Scholar 

  33. 33

    Hazlett, E. L., Zhang, Y., Stites, R. W. & O’Hara, K. M. Realization of a resonant Fermi gas with a large effective range. Phys. Rev. Lett. 108, 045304 (2012).

    ADS  Google Scholar 

  34. 34

    Kohstall, C., Zaccanti, M., Jag, M. & Trenkwalder, A. Metastability and coherence of repulsive polarons in a strongly interacting Fermi mixture. Nature 485, 615–618 (2012).

    ADS  Google Scholar 

  35. 35

    Gaebler, J. P., Stewart, J. T., Bohn, J. L. & Jin, D. S. p-wave Feshbach molecules. Phys. Rev. Lett. 98, 200403 (2007).

    ADS  Google Scholar 

  36. 36

    Chin, C. & Julienne, P. S. Radio-frequency transitions on weakly bound ultracold molecules. Phys. Rev. A 71, 012713 (2005).

    ADS  Google Scholar 

  37. 37

    Pieri, P., Perali, A. & Strinati, G. C. Enhanced paraconductivity-like fluctuations in the radiofrequency spectra of ultracold Fermi atoms. Nature Phys. 5, 736–740 (2009).

    ADS  Google Scholar 

  38. 38

    Schneider, W. & Randeria, M. Universal short-distance structure of the single-particle spectral function of dilute Fermi gases. Phys. Rev. A 81, 021601 (2010).

    ADS  Google Scholar 

  39. 39

    Braaten, E., Kang, D. & Platter, L. Short-time operator product expansion for rf spectroscopy of a strongly interacting Fermi gas. Phys. Rev. Lett. 104, 223004 (2010).

    ADS  Google Scholar 

  40. 40

    Stewart, J. T., Gaebler, J. P., Drake, T. E. & Jin, D. S. Verification of universal relations in a strongly interacting Fermi gas. Phys. Rev. Lett. 104, 235301 (2010).

    ADS  Google Scholar 

  41. 41

    Chevy, F. et al. Resonant scattering properties close to a p-wave Feshbach resonance. Phys. Rev. A 71, 062710 (2005).

    ADS  Google Scholar 

  42. 42

    Inada, Y. et al. Collisional properties of p-wave Feshbach molecules. Phys. Rev. Lett. 101, 100401 (2008).

    ADS  Google Scholar 

  43. 43

    Nakasuji, T., Yoshida, J. & Mukaiyama, T. Experimental determination of p-wave scattering parameters in ultracold 6Li atoms. Phys. Rev. A 88, 012710 (2013).

    ADS  Google Scholar 

  44. 44

    Ohashi, Y. BCS–BEC crossover in a gas of Fermi atoms with a p-wave Feshbach resonance. Phys. Rev. Lett. 94, 050403 (2005).

    ADS  Google Scholar 

  45. 45

    Gurarie, V. & Radzihovsky, L. Resonantly paired fermionic superfluids. Ann. Phys. 322, 2–119 (2007).

    ADS  MathSciNet  MATH  Google Scholar 

  46. 46

    Sagi, Y., Drake, T. E., Paudel, R. & Jin, D. S. Measurement of the homogeneous contact of a unitary Fermi gas. Phys. Rev. Lett. 109, 220402 (2012).

    ADS  Google Scholar 

  47. 47

    Pricoupenko, L. Modeling interactions for resonant p-wave scattering. Phys. Rev. Lett. 96, 050401 (2006).

    ADS  Google Scholar 

  48. 48

    Shenoy, V. B. & Ho, T.-L. Nature and properties of a repulsive Fermi gas in the upper branch of the energy spectrum. Phys. Rev. Lett. 107, 210401 (2011).

    ADS  Google Scholar 

  49. 49

    Fuchs, J. et al. Binding energies of 6Li p-wave Feshbach molecules. Phys. Rev. A 77, 053616 (2008).

    ADS  Google Scholar 

  50. 50

    Gubbels, K. B. & Stoof, H. T. C. Theory for p-wave Feshbach molecules. Phys. Rev. Lett. 99, 190406 (2007).

    ADS  Google Scholar 

  51. 51

    Jin, D. S., Gaebler, J. P. & Stewart, J. T. in Proceedings of the XVIII International Conference on Laser Spectroscopy (eds Hollberg, L., Bergquist, J. & Kasevich, M.) 127–137 (World Scientific, 2008).

    Google Scholar 

  52. 52

    Partridge, G. B., Strecker, K. E., Kamar, R. I., Jack, M. W. & Hulet, R. G. Molecular probe of pairing in the BEC–BCS crossover. Phys. Rev. Lett. 95, 020404 (2005).

    ADS  Google Scholar 

  53. 53

    Kuhnle, E. D. et al. Universal behavior of pair correlations in a strongly interacting Fermi gas. Phys. Rev. Lett. 105, 070402 (2010).

    ADS  Google Scholar 

  54. 54

    Navon, N., Nascimbène, S., Chevy, F. & Salomon, C. The equation of state of a low-temperature Fermi gas with tunable interactions. Science 328, 729–732 (2010).

    ADS  Google Scholar 

  55. 55

    Kuhnle, E. D. et al. Temperature dependence of the universal contact parameter in a unitary Fermi gas. Phys. Rev. Lett. 106, 170402 (2011).

    ADS  Google Scholar 

  56. 56

    Levinsen, J., Cooper, N. R. & Gurarie, V. Stability of fermionic gases close to a p-wave Feshbach resonance. Phys. Rev. A 78, 063616 (2008).

    ADS  Google Scholar 

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We thank F. Chevy for discussion and shared notes concerning F versus δBm. We also thank N. Zuber for experimental assistance, and J. Bohn, B. Ruzic, S. Tan, E. Taylor, P. Zhang and Q. Zhou for discussion. This work was supported by AFOSR under FA9550-13-1-0063, ARO under W911NF-15-1-0603, the Croucher Foundation, RGC under under 17306414, NKBRSFC, NSERC and NSFC under 11474179, and the Tsinghua University Initiative Scientific Research Program.

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C.L., S.S. and S.T. performed the experiments. C.L., S.S., S.T. and J.H.T. analysed the data. All authors contributed to the interpretation of the spectra. Z.Y., S.T. and S.Z. developed the two-channel model of the dynamics. All authors contributed to the preparation of the manuscript.

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Correspondence to Shizhong Zhang or Joseph H. Thywissen.

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The authors declare no competing financial interests.

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Luciuk, C., Trotzky, S., Smale, S. et al. Evidence for universal relations describing a gas with p-wave interactions. Nature Phys 12, 599–605 (2016). https://doi.org/10.1038/nphys3670

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