Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Unitary p-wave interactions between fermions in an optical lattice


Exchange-antisymmetric pair wavefunctions in fermionic systems can give rise to unconventional superconductors and superfluids1,2,3. The realization of these states in controllable quantum systems, such as ultracold gases, could enable new types of quantum simulations4,5,6,7,8, topological quantum gates9,10,11 and exotic few-body states12,13,14,15. However, p-wave and other antisymmetric interactions are weak in naturally occurring systems16,17, and their enhancement via Feshbach resonances in ultracold systems has been limited by three-body loss18,19,20,21,22,23,24. Here we create isolated pairs of spin-polarized fermionic atoms in a multiorbital three-dimensional optical lattice. We spectroscopically measure elastic p-wave interaction energies of strongly interacting pairs of atoms near a magnetic Feshbach resonance. The interaction strengths are widely tunable by the magnetic field and confinement strength, and yet collapse onto a universal curve when rescaled by the harmonic energy and length scales of a single lattice site. The absence of three-body processes enables the observation of elastic unitary p-wave interactions, as well as coherent oscillations between free-atom and interacting-pair states. All observations are compared both to an exact solution using a p-wave pseudopotential and to numerical solutions using an ab initio interaction potential. The understanding and control of on-site p-wave interactions provides a necessary component for the assembly of multiorbital lattice models25,26 and a starting point for investigations of how to protect such systems from three-body recombination in the presence of tunnelling, for instance using Pauli blocking and lattice engineering27,28.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Get just this article for as long as you need it


Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Spectroscopy of p-wave interactions between spin-polarized fermions.
Fig. 2: Characterization of unitary and elastic p-wave interactions.
Fig. 3: Coherent manipulation of p-wave interacting pairs.
Fig. 4: Lifetime of p-wave interacting pairs.

Data availability

Source data are provided with this paper.


  1. Volovik, G. The Universe in a Helium Droplet, International Series of Monographs on Physics (Clarendon Press, 2003).

  2. Ivanov, D. A. Non-Abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86, 268–271 (2001).

    Article  ADS  CAS  Google Scholar 

  3. Mizushima, T. et al. Symmetry-protected topological superfluids and superconductors: from the basics to 3He. J. Phys. Soc. Jpn 85, 022001 (2016).

  4. Botelho, S. S. & Sá de Melo, C. A. R. Quantum phase transition in the BCS-to-BEC evolution of p-wave Fermi gases. J. Low Temp. Phys. 140, 409–428 (2005).

    Article  ADS  CAS  Google Scholar 

  5. Gurarie, V., Radzihovsky, L. & Andreev, A. V. Quantum phase transitions across a p-wave Feshbach resonance. Phys. Rev. Lett. 94, 230403 (2005).

    Article  ADS  CAS  Google Scholar 

  6. Cheng, C.-H. & Yip, S.-K. Anisotropic Fermi superfluid via p-wave Feshbach resonance. Phys. Rev. Lett. 95, 070404 (2005).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  8. Zhao, E. & Liu, W. V. Orbital order in Mott insulators of spinless p-band fermions. Phys. Rev. Lett. 100, 160403 (2008).

    Article  ADS  Google Scholar 

  9. Tewari, S., Das Sarma, S., Nayak, C., Zhang, C. & Zoller, P. Quantum computation using vortices and majorana zero modes of a px + ipy superfluid of fermionic cold atoms. Phys. Rev. Lett. 98, 010506 (2007).

    Article  ADS  Google Scholar 

  10. Zhang, C., Tewari, S. & Das Sarma, S. Bell’s inequality and universal quantum gates in a cold-atom chiral fermionic p-wave superfluid. Phys. Rev. Lett. 99, 220502 (2007).

    Article  ADS  Google Scholar 

  11. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  13. D’Incao, J. P., Esry, B. D. & Greene, C. H. Ultracold atom-molecule collisions with fermionic atoms. Phys. Rev. A 77, 052709 (2008).

    Article  ADS  Google Scholar 

  14. Nishida, Y., Moroz, S. & Son, D. T. Super Efimov effect of resonantly interacting fermions in two dimensions. Phys. Rev. Lett. 110, 235301 (2013).

    Article  ADS  Google Scholar 

  15. Wang, Y., D’Incao, J. P. & Greene, C. H. Universal three-body physics for fermionic dipoles. Phys. Rev. Lett. 107, 233201 (2011).

    Article  ADS  Google Scholar 

  16. Martin, M. J. et al. A quantum many-body spin system in an optical lattice clock. Science 341, 632–636 (2013).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  17. Lemke, N. D. et al. p-wave cold collisions in an optical lattice clock. Phys. Rev. Lett. 107, 103902 (2011).

    Article  ADS  CAS  Google Scholar 

  18. 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).

    Article  ADS  CAS  Google Scholar 

  19. Suno, H., Esry, B. D. & Greene, C. H. Recombination of three ultracold fermionic atoms. Phys. Rev. Lett. 90, 053202 (2003).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  24. Waseem, M., Yoshida, J., Saito, T. & Mukaiyama, T. Unitarity-limited behavior of three-body collisions in a p-wave interacting Fermi gas. Phys. Rev. A 98, 020702 (2018).

    Article  ADS  CAS  Google Scholar 

  25. Li, X. & Liu, W. V. Physics of higher orbital bands in optical lattices: a review. Rep. Prog. Phys. 79, 116401 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  26. Dutta, O. et al. Non-standard Hubbard models in optical lattices: a review. Rep. Prog. Phys. 78, 066001 (2015).

    Article  ADS  Google Scholar 

  27. Han, Y.-J. et al. Stabilization of the p-wave superfluid state in an optical lattice. Phys. Rev. Lett. 103, 070404 (2009).

    Article  ADS  Google Scholar 

  28. Fedorov, A. K., Yudson, V. I. & Shlyapnikov, G. V. p-wave superfluidity of atomic lattice fermions. Phys. Rev. A 95, 043615 (2017).

    Article  ADS  Google Scholar 

  29. Alicea, J. New directions in the pursuit of Majorana fermions in solid state systems. Rep. Prog. Phys. 75, 076501 (2012).

    Article  ADS  Google Scholar 

  30. Chang, Y.-T., Senaratne, R., Cavazos-Cavazos, D. & Hulet, R. G. Collisional loss of one-dimensional fermions near a p-wave Feshbach resonance. Phys. Rev. Lett. 125, 263402 (2020).

    Article  ADS  CAS  Google Scholar 

  31. Marcum, A. S., Fonta, F. R., Mawardi Ismail, A. & O’Hara, K. M. Suppression of three-body loss near a p-wave resonance due to quasi-1D confinement. Preprint at (2020).

  32. Idziaszek, Z. Analytical solutions for two atoms in a harmonic trap: p-wave interactions. Phys. Rev. A 79, 062701 (2009).

    Article  ADS  Google Scholar 

  33. Kanjilal, K. & Blume, D. Nondivergent pseudopotential treatment of spin-polarized fermions under one- and three-dimensional harmonic confinement. Phys. Rev. A 70, 042709 (2004).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  35. Stöferle, T., Moritz, H., Günter, K., Köhl, M. & Esslinger, T. Molecules of fermionic atoms in an optical lattice. Phys. Rev. Lett. 96, 030401 (2006).

    Article  ADS  Google Scholar 

  36. Hartke, T., Oreg, B., Jia, N. & Zwierlein, M. Quantum register of fermion pairs. Nature 601, 537–541 (2022).

    Article  ADS  CAS  Google Scholar 

  37. Müller, T., Fölling, S., Widera, A. & Bloch, I. State preparation and dynamics of ultracold atoms in higher lattice orbitals. Phys. Rev. Lett. 99, 200405 (2007).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  39. Busch, T., Englert, B.-G., Rzażewski, K. & Wilkens, M. Two cold atoms in a harmonic trap. Found. Phys. 28, 549–559 (1998).

    Article  Google Scholar 

  40. 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).

    Article  ADS  Google Scholar 

  41. Ahmed-Braun, D. J. M. et al. Probing open- and closed-channel p-wave resonances. Phys. Rev. Res. 3, 033269 (2021).

    Article  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

  43. Foster, M. S., Gurarie, V., Dzero, M. & Yuzbashyan, E. A. Quench-induced floquet topological p-wave superfluids. Phys. Rev. Lett. 113, 076403 (2014).

    Article  ADS  CAS  Google Scholar 

  44. Tokura, Y. & Nagaosa, N. Orbital physics in transition-metal oxides. Science 288, 462–468 (2000).

    Article  ADS  CAS  Google Scholar 

  45. Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).

    Article  ADS  CAS  Google Scholar 

  46. Mamaev, M. et al. Collective p-wave orbital dynamics of ultracold fermions. Phys. Rev. Lett. 127, 143401 (2021).

    Article  ADS  CAS  Google Scholar 

  47. Bakr, W. S. et al. Orbital excitation blockade and algorithmic cooling in quantum gases. Nature 480, 500–503 (2011).

    Article  ADS  CAS  Google Scholar 

  48. Mamaev, M., Thywissen, J. H. & Rey, A. M. Quantum computation toolbox for decoherence-free qubits using multi-band alkali atoms. Adv. Quantum Technol. 3, 1900132 (2020).

    Article  CAS  Google Scholar 

  49. Marco, L. D. et al. A degenerate Fermi gas of polar molecules. Science 363, 853–856 (2019).

    Article  ADS  Google Scholar 

  50. Duda, M. et al. Long-lived fermionic Feshbach molecules with tunable p-wave interactions. Preprint at (2022).

  51. Edge, G. J. A. et al. Imaging and addressing of individual fermionic atoms in an optical lattice. Phys. Rev. A 92, 063406 (2015).

    Article  ADS  Google Scholar 

  52. Anderson, R. et al. Conductivity spectrum of ultracold atoms in an optical lattice. Phys. Rev. Lett. 122, 153602 (2019).

    Article  ADS  CAS  Google Scholar 

  53. Omont, A. On the theory of collisions of atoms in Rydberg states with neutral particles. J. Phys. France 38, 1343–1359 (1977).

    Article  CAS  Google Scholar 

  54. Mentink, J. & Kokkelmans, S. Two interacting atoms in an optical lattice site with anharmonic terms. Phys. Rev. A 79, 032709 (2009).

    Article  ADS  Google Scholar 

Download references


We acknowledge insightful discussions with F. Chevy and S. Zhang and helpful manuscript comments from J. Bohn, A. Kaufman and R. Learn. This work is supported by the AFOSR grants FA9550-19-1-0275, FA9550-19-1-7044 and FA9550-19-1-0365, by ARO W911NF-15-1-0603, by the NSF’s JILA-PFC PHY-1734006 and PHY-2012125 grants, by NIST and by NSERC.

Author information

Authors and Affiliations



V.V., P.X., F.C., C.J.F. and J.H.T. developed the experimental techniques and collected the data. V.V., P.X., M.M., F.C. and C.J.F. performed the data analysis. M.M., T.B., C.J.F., J.P.D’I. and A.M.R. developed the theoretical models. All authors contributed to writing the manuscript.

Corresponding authors

Correspondence to Cora J. Fujiwara, Ana Maria Rey or Joseph H. Thywissen.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Xiaopeng Li and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

This file contains Supplementary text, equations and references.

Source data

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Venu, V., Xu, P., Mamaev, M. et al. Unitary p-wave interactions between fermions in an optical lattice. Nature 613, 262–267 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing