Pairing-gap discrepancy and Goldstone-mode–pair-breaking coupling in the BEC-BCS crossover of a homogeneous ultracold Fermi gas

Momentum-resolved Bragg spectroscopy of a homogeneous, strongly interacting two-component Fermi gas (Biss et al., Phys. Rev. Lett. 128, 100401 (2022)) mapped the low-energy dynamic structure factor $S(q,\omega)$ across the entire BEC-BCS crossover and extracted the pairing gap $\Delta(1/(k_F a))$ from the pair-breaking threshold. The result is a two-front theory-experiment discrepancy that remains unresolved: no microscopic treatment — neither zero-temperature self-consistent T-matrix theory nor strong-coupling theory including Gor'kov-Melik-Barkhudarov particle-hole screening — reproduces the measured gap on both the BCS and the BEC side simultaneously; and the observed coupling of the Goldstone (Bogoliubov-Anderson) mode to the pair-breaking continuum, including the curvature of the collective branch, lacks a quantitative microscopic account, with the sign of the zero-temperature curvature at unitarity contested in the published literature. The problem is to identify the physical mechanism behind the gap reduction, construct a theory valid at the experimentally realized entropy that matches all published benchmarks, and pin down the mode-continuum coupling, supported by entropy/temperature control sharp enough to discriminate mechanisms.

Background

A two-component ultracold gas of fermionic atoms (here $^6$Li) near a Feshbach resonance allows the $s$-wave scattering length $a$, and hence the interaction strength, to be tuned at fixed density. The dimensionless parameter $1/(k_F a)$, with $k_F$ the Fermi wave vector, spans the BEC-BCS crossover: for $1/(k_F a)\gg 1$ fermions bind into tightly bound molecules that form a Bose-Einstein condensate (BEC), while for $1/(k_F a)\ll -1$ they form weakly bound Cooper pairs in a Bardeen-Cooper-Schrieffer (BCS) superfluid; at $1/(k_F a)=0$ (unitarity) the scattering length diverges and the system is maximally strongly coupled. A fermionic superfluid has two kinds of low-energy excitations: single-particle (particle-hole) excitations, which require breaking Cooper pairs and therefore cost at least twice the pairing gap $\Delta$, and a collective phase mode of the order parameter — the Bogoliubov-Anderson or Goldstone mode, an acoustic phonon at small momentum. Both appear in the dynamic structure factor $S(q,\omega)$, the probability for the system to absorb energy $\hbar\omega$ and momentum $\hbar q$. Bragg spectroscopy measures $S(q,\omega)$ by stimulating two-photon scattering from the gas; performing it in a homogeneous box trap (rather than a harmonic trap) removes density inhomogeneity and allows direct comparison with theory for the uniform system. In this setting Biss et al. (Phys. Rev. Lett. 128, 100401 (2022)) measured the momentum-resolved $S(q,\omega)$ of a homogeneous $^6$Li gas from the deep BEC to the BCS regime, observed the phonon branch bending down as it approaches the pair-breaking continuum in an avoided-crossing-like fashion (a coupling between the Goldstone mode and pair breaking that had been predicted, e.g. by Combescot, Kagan, and Stringari, but not previously observed), fitted the phonon dispersion $\omega_q = v_s q(1+\zeta q^2)$ to extract the speed of sound $v_s$ and curvature $\zeta$, and extracted $\Delta$ from the threshold of the pair-breaking continuum, which for positive chemical potential $\mu$ (with $k_\mu=\sqrt{2m\mu/\hbar}$) sets in at an energy $\omega = 2\Delta$ for momentum transfers $q<2k_\mu$. Their gap data agree with earlier measurements (Schirotzek et al. 2008; Hoinka et al. 2017) but disagree with theory in a structured way: the zero-temperature self-consistent T-matrix calculation of Haussmann, Rantner, Cerrito, and Zwerger — a scheme that resums particle-particle (pairing) fluctuations self-consistently — agrees in the BEC and crossover regimes but lies significantly above the data on the BCS side; a finite-temperature explanation is disfavored because the gas stays at nearly constant entropy while the interaction is ramped, whereas matching the finite-entropy T-matrix to the data would require an entropy that varies far more than observed; and the strong-coupling calculation of Pisani, Pieri, and Strinati that includes particle-hole correlations — the physics behind the Gor'kov-Melik-Barkhudarov (GMB) correction, the famous screening-induced reduction of the pairing gap and critical temperature in the weak-coupling limit — agrees on the BCS side but lies significantly above the data on the BEC side. Neither theory is consistent with the entire dataset, and the spectra themselves are compared only to a quasiparticle random-phase approximation (QRPA, a linear-response treatment built on mean-field quasiparticles with the gap supplied as input), not to a self-contained microscopic theory. A further subtlety, raised by Castin's published Comment, is that the experimental curvature $\zeta=-0.085(8)/k_F^2$ at unitarity was extracted at fairly high momenta and temperature $T\approx 0.13 T_F\approx 0.8 T_c$, where fitting bias and thermal shifts of order $-0.1$ in units of $1/k_F^2$ can invert the sign of the zero-temperature, low-momentum curvature — a sign that determines the dominant low-temperature phonon damping mechanism (the well-studied three-phonon Beliaev process if $\zeta>0$, or the still-unobserved four-phonon Landau-Khalatnikov process if $\zeta<0$). Finally, the temperature of a homogeneous gas in the crossover is hard to measure directly; the entropy per particle $S/N$ is the invariant that is approximately conserved across interaction ramps, and its experimental determination is a limiting factor in comparing finite-temperature theory to the data.

Problem Statement

For the balanced, homogeneous, strongly interacting two-component Fermi gas realized with $^6$Li atoms near a Feshbach resonance and probed by momentum-resolved Bragg spectroscopy in a box trap: (i) determine the microscopic origin of the measured reduction of the pairing gap $\Delta$ on the BCS side of the crossover relative to zero-temperature self-consistent T-matrix predictions — discriminating between finite-temperature/entropy effects, particle-hole (Gor'kov-Melik-Barkhudarov) screening, and other strong-coupling approximation errors — and construct a theoretical treatment, evaluated at the experimentally realized near-constant entropy, that reproduces the measured $\Delta(1/(k_F a))$ within experimental uncertainties on the BCS and the BEC side simultaneously, where current GMB-including calculations overestimate the data; (ii) provide the quantitative microscopic account of the coupling between the Goldstone (Bogoliubov-Anderson) mode and the pair-breaking continuum that reproduces the observed avoided-crossing-like downbending of the collective branch, its dependence on interaction strength $1/(k_F a)$ and momentum $q$, and the curvature parameter $\zeta$, including the sign and magnitude of the zero-temperature low-momentum curvature at unitarity; and (iii) specify the entropy and temperature determination across interaction ramps (improved thermometry or independent entropy measurement): state the precision that would discriminate the candidate mechanisms and design a measurement protocol that reaches it. These deliverables share one overarching objective: a quantitatively validated microscopic description of the low-energy excitation spectrum of the homogeneous Fermi gas across the entire BEC-BCS crossover, consistent with the published Bragg benchmarks. The task is resolved when the theoretical deliverables (i) and (ii) are both met, together with deliverable (iii) as the specified precision and the corresponding measurement protocol; executing the measurement is not required of the solver.

References

1. Biss, H., Sobirey, L., Luick, N., Bohlen, M., Kinnunen, J. J., Bruun, G. M., Lompe, T., and Moritz, H. (2022). Excitation Spectrum and Superfluid Gap of an Ultracold Fermi Gas. Physical Review Letters 128, 100401. DOI: 10.1103/PhysRevLett.128.100401. https://arxiv.org/abs/2105.09820
2. Haussmann, R., Rantner, W., Cerrito, S., and Zwerger, W. (2007). Thermodynamics of the BCS-BEC crossover. Physical Review A 75, 023610. DOI: 10.1103/PhysRevA.75.023610. https://doi.org/10.1103/PhysRevA.75.023610
3. Pisani, L., Pieri, P., and Strinati, G. C. (2018). Gap equation with pairing correlations beyond the mean-field approximation and its equivalence to a Hugenholtz-Pines condition for fermion pairs. Physical Review B 98, 104507. DOI: 10.1103/PhysRevB.98.104507. https://doi.org/10.1103/PhysRevB.98.104507
4. Schirotzek, A., Shin, Y.-i., Schunck, C. H., and Ketterle, W. (2008). Determination of the Superfluid Gap in Atomic Fermi Gases by Quasiparticle Spectroscopy. Physical Review Letters 101, 140403. DOI: 10.1103/PhysRevLett.101.140403. https://doi.org/10.1103/PhysRevLett.101.140403
5. Hoinka, S., Dyke, P., Lingham, M. G., Kinnunen, J. J., Bruun, G. M., and Vale, C. J. (2017). Goldstone mode and pair-breaking excitations in atomic Fermi superfluids. Nature Physics 13, 943-946. DOI: 10.1038/nphys4187. https://doi.org/10.1038/nphys4187
6. Combescot, R., Kagan, M. Yu., and Stringari, S. (2006). Collective mode of homogeneous superfluid Fermi gases in the BEC-BCS crossover. Physical Review A 74, 042717. DOI: 10.1103/PhysRevA.74.042717. https://doi.org/10.1103/PhysRevA.74.042717
7. Kurkjian, H., Castin, Y., and Sinatra, A. (2016). Concavity of the collective excitation branch of a Fermi gas in the BEC-BCS crossover. Physical Review A 93, 013623. DOI: 10.1103/PhysRevA.93.013623. https://doi.org/10.1103/PhysRevA.93.013623
8. Vale, C. J., and Zwierlein, M. (2021). Spectroscopic probes of quantum gases. Nature Physics 17, 1305-1315. DOI: 10.1038/s41567-021-01434-6. https://doi.org/10.1038/s41567-021-01434-6
9. Castin, Y. (2024). Comment on "Excitation Spectrum and Superfluid Gap of an Ultracold Fermi Gas". Physical Review Letters 133, 109301. DOI: 10.1103/PhysRevLett.133.109301. https://arxiv.org/abs/2312.07994
10. Biss, H., Sobirey, L., Luick, N., Bohlen, M., Kinnunen, J. J., Bruun, G. M., Lompe, T., and Moritz, H. (2024). Biss et al. Reply. Physical Review Letters 133, 109302. DOI: 10.1103/PhysRevLett.133.109302. https://doi.org/10.1103/PhysRevLett.133.109302
11. Castin, Y. (2025). Questions ouvertes pour les gaz de fermions en interaction forte et de portée nulle. Comptes Rendus. Physique 26, 393-461. DOI: 10.5802/crphys.244. https://doi.org/10.5802/crphys.244
12. Calvanese Strinati, G. (2023). Recent advances in the theory of the BCS-BEC crossover for fermionic superfluidity. Physica C: Superconductivity and its Applications 614, 1354377. DOI: 10.1016/j.physc.2023.1354377. https://arxiv.org/abs/2403.12515
13. Faigle-Cedzich, B. M., Pawlowski, J. M., and Wetterich, C. (2023). Towards quantitative precision in ultracold atoms with functional renormalisation. arXiv:2307.14787. https://arxiv.org/abs/2307.14787
14. Li, X., Wang, S., Luo, X., Zhou, Y.-Y., Xie, K., Shen, H.-C., Nie, Y.-Z., Chen, Q., Hu, H., Chen, Y.-A., Yao, X.-C., and Pan, J.-W. (2024). Observation and quantification of the pseudogap in unitary Fermi gases. Nature 626, 288-293. DOI: 10.1038/s41586-023-06964-y. https://doi.org/10.1038/s41586-023-06964-y
15. Link, M., Gao, K., Kell, A., Breyer, M., Eberz, D., Rauf, B., and Köhl, M. (2023). Machine Learning the Phase Diagram of a Strongly Interacting Fermi Gas. Physical Review Letters 130, 203401. DOI: 10.1103/PhysRevLett.130.203401. https://doi.org/10.1103/PhysRevLett.130.203401
16. Yan, Z., Patel, P. B., Mukherjee, B., Vale, C. J., Fletcher, R. J., and Zwierlein, M. W. (2024). Thermography of the superfluid transition in a strongly interacting Fermi gas. Science 383, 629-633. DOI: 10.1126/science.adg3430. https://doi.org/10.1126/science.adg3430
17. Urban, M., and Ramanan, S. (2026). Hartree shift and pairing gap in ultracold Fermi gases in the framework of low-momentum interactions. arXiv:2602.17420. https://arxiv.org/abs/2602.17420