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arXiv:2509.04738 · Physical Review Letters

Temperature Dependence of p -Wave Contacts in a Harmonically Trapped Fermi Gas

Kenta Nagase, Hikaru Takahashi, Soki Oshima, Takashi Mukaiyama

Published Dec 2, 2025• 3 citations
arXivHTMLPDF

Abstract

We study the dependence of the p-wave contact on the Fermi temperature T_{F} and reduced temperature T/T_{F} based on the number of closed-channel molecules. From the anisotropic pattern of dissociated molecules, we resolve the narrow m=0 and m=±1 dipolar splitting of the p-wave Feshbach resonance in ^{6}Li, enabling the independent determination of the contact for all three m components. For each component, we identify a near-resonant scaling with sqrt[T_{F}], indicating the contribution of the normalized effective range k_{F}R_{e}. In addition, we show that the peak contacts observed near resonance increase as T/T_{F} is lowered, a trend that is accurately captured by estimates based on the second-order virial expansion. Our results, together with estimates from the p-wave virial expansion, provide a route toward a complete understanding of the thermodynamics of resonantly enhanced p-wave Fermi gases.

DOI
10.1103/3mxj-ym4b

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We study the dependence of the p-wave contact on the Fermi temperature T_{F} and reduced temperature T/T_{F} based on the number of closed-channel molecules. From the anisotropic pattern of dissociated molecules, we resolve the narrow m=0 and m=±1 dipolar splitting of the p-wave Feshbach resonance in ^{6}Li, enabling the independent determination of the contact for all three m components. For each component, we identify a near-resonant scaling with sqrt[T_{F}], indicating the contribution of the normalized effective range k_{F}R_{e}. In addition, we show that the peak contacts observed near resonance increase as T/T_{F} is lowered, a trend that is accurately captured by estimates based on the second-order virial expansion. Our results, together with estimates from the p-wave virial expansion, provide a route toward a complete understanding of the thermodynamics of resonantly enhanced p-wave Fermi gases.
Using radio-frequency (rf) spectroscopy, we measure the ppp-wave contacts in a Fermi gas of 6Li^{6}\mathrm{Li}6Li atoms near the ppp-wave Feshbach resonance. The rf spectrum exhibits clear asymptotic behavior, characterized by ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}^{\ensuremath{-}1/2}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}^{\ensuremath{-}3/2}$ dependencies. The magnetic-field dependence of the ppp-wave contacts agrees reasonably well with the virial expansion theory in a detuned magnetic-field range, validating the theory near half of the Fermi temperature. The ppp-wave contacts measured in this article constitute a second data set, complementing the data obtained from the 40K^{40}\mathrm{K}40K system and contributing valuable insights into ppp-wave interactions in ultracold Fermi gases.
We theoretically investigate the high-temperature thermodynamics of a strongly interacting trapped Fermi gas near either sss-wave or ppp-wave Feshbach resonances, using a second-order quantum virial expansion. The second virial coefficient is calculated based on the energy spectrum of two interacting fermions in a harmonic trap. We consider both isotropic and anisotropic harmonic potentials. For the two-fermion interaction, either sss-wave or ppp-wave, we use a pseudopotential parametrized by a scattering length and an effective range. This turns out to be the simplest way of encoding the energy dependence of the low-energy scattering amplitude or phase shift. This treatment of the pseudopotential can be easily generalized to higher partial-wave interactions. We discuss how the second virial coefficient and thermodynamics are affected by the existence of these finite-range interaction effects. The virial expansion result for a strongly interacting sss-wave Fermi gas has already been proved very useful. In the case of ppp-wave interactions, our results for the high-temperature equation of state are applicable to future high-precision thermodynamic measurements for a spin-polarized Fermi gas near a ppp-wave Feshbach resonance.
We theoretically investigate the high-temperature thermodynamics of a harmonically trapped Fermi gas across a narrow Feshbach resonance by using the second-order quantum virial expansion, and point out some new features compared to the broad resonance. The interatomic interaction is modeled by the pseudopotential with an additional parameter, i.e., the effective range, to characterize the narrow-resonance width. Deep inside the width of a narrow Feshbach resonance, we find the second virial coefficient evolves with the effective range from the well-known universal value 1/41/41/4 in the broad-resonance limit to one another value 1/21/21/2 in the narrow-resonance limit. This means the Fermi gas interacts more strongly at the narrow resonance. In addition, far beyond the resonance width, we find the harmonically trapped Fermi gas still manifests an appreciable interaction effect across a narrow Feshbach resonance, which is contrary to our knowledge of the broad Feshbach resonance. All our results can be directly tested in current narrow Feshbach resonance experiments, which are generally carried out in a harmonic trap.
Motivated by the experimental realization of single-component degenerate Fermi gases of polar ground state KRb molecules with intrinsic two-body losses [L. De Marco et al., A degenerate Fermi gas of polar molecules, Science 363, 853 (2019).SCIEAS0036-807510.1126/science.aau7230], this work studies the finite-temperature loss rate of single-component Fermi gases with weak interactions. First, we establish a relationship between the two-body loss rate and the p-wave contact. Second, we evaluate the contact of the homogeneous system in the low-temperature regime using p-wave Fermi liquid theory and in the high-temperature regime using the second-order virial expansion. Third, conjecturing that there are no phase transitions between the two temperature regimes, we smoothly interpolate the results to intermediate temperatures. It is found that the contact is constant at temperatures close to zero and increases first quadratically with increasing temperature and finally-in agreement with the Bethe-Wigner threshold law-linearly at high temperatures. Fourth, applying the local-density approximation, we obtain the loss-rate coefficient for the harmonically trapped system, reproducing the experimental KRb loss measurements within a unified theoretical framework over a wide temperature regime without fitting parameters. Our results for the contact are not only applicable to molecular p-wave gases but also to atomic single-component Fermi gases, such as ^{40}K and ^{6}Li.
Using a second-order quantum virial expansion, we theoretically study the high-temperature thermodynamics of spin-polarized Fermi gases with ppp-wave interaction confined in two-dimensional harmonic traps. The second virial coefficient is calculated based on the energy spectrum of two harmonically trapped fermions. We find that the two-body spectrum can correctly be evaluated only when the ppp-wave effective range is properly considered, which is referred to as the well-known Wigner's bound. According to the quantum virial expansion, the high-temperature thermodynamics of the system is then discussed based on the two-body solutions. In addition, we investigate the high-temperature behavior of ppp-wave contacts in harmonic traps within the quantum virial expansion, which characterize a set of universal relations of strongly interacting spin-polarized Fermi gases. Our results can provide an accurate benchmark at high temperature for future numerical simulations as well as high-precision thermodynamic measurements of harmonically trapped spin-polarized Fermi gases in two dimensions.
We investigate universal relations in a spinless Fermi gas near a p-wave Feshbach resonance, and show that the momentum distribution n_{k} has an asymptote proportional to k^{-2} with the proportionality constant-the p-wave contact-scaling with the number of closed-channel molecules. We prove the adiabatic sweep theorem for a p-wave resonance which reveals the thermodynamic implication of the p-wave contact. In contrast to the unitary Fermi gas in which Tan's contact is universal, the p-wave contact depends on the short-range details of the interaction.