Trapping-potential dependence of the unitary Fermi gas at the BCS-BEC crossover

TL;DR

This work develops a systematic effective-field-theory framework to understand how trapping potentials modify the unitary Fermi gas across the BCS–BEC crossover. By combining a superfluid EFT for the Goldstone phonon with a WKB expansion (in the trap gradient) and a large-charge expansion, the authors show how the trap introduces three scales (μ, q0, varpi) and define a double expansion in η = q0/μ and δ = varpi/q0, with ε = varpi/μ controlling the large-density limit. They demonstrate that, in a trapped geometry, the fluctuation spectrum becomes discrete and can be analyzed via a self-adjoint Sturm–Liouville problem at LO (and a higher-order generalization at NLO), while the dynamic structure factor connects spectral information to measurable response functions. The key findings are that the spectrum density grows for steeper traps and approaches equidistant levels for flatter potentials (approaching the spherical box limit), with low-energy dispersion corrections potentially concave and high-energy corrections governed by low-energy EFT parameters. Overall, the paper provides a quantitative, gradient-based EFT method to interpret trapped Fermi-gas experiments and connect observed excitations to the underlying homogeneous-unitary theory.

Abstract

Cold-atom experiments which measure Fermi-gas properties near unitarity confine fermionic atoms to a region of space using trapping potentials of various shapes. The presence of a trapping potential introduces a new characteristic physical scale in the superfluid EFT which, inter alia, describes the acoustic branch of excitations in the far infrared well below the scale of the superfluid gap. In this EFT there is a clear hierarchy of scales, and corrections to the homogeneous system due to the trapping potential may be organized into three regions with distinct power counting that relies on both the EFT derivative expansion, and the WKB approximation, which is an expansion in gradients of the trapping potential. The energy spectrum of the superfluid system is obtained in each of the regions by explicit computation of the phonon-field fluctuations, and by the modifications to the dynamic structure factor due to the corresponding density fluctuations. The most significant deviations from linear dispersion due to the trapping potential are found in the far infrared region of the superfluid EFT.

Figures (2)

• Figure 1: Fit of DMC simulation data over the range $Q=8-80$, as described in the text. The solid black line is leading order in the large-charge expansion, and the gray band is the nlo fit, including the Casimir correction. The simulation data are as described in the text.
• Figure 2: Scale separation for the fluctuation energy $q_0$. The dynamics is controlled by the low-energy scale $\varpi$, fixed by the confining potential and the chemical potential $\mu$, acting as a high-energy scale. In the intermediate region ($q_0 = \mathcal{O}\pqty{\sqrt{\varpi \mu}}$), the physics is well described by the physical optics approximation for the lo eft and the measured dispersion is linear. At lower energies $\varpi \ll q_0 \ll \sqrt{\varpi \mu}$, higher orders in the wkb expansion are needed. At higher energies $\sqrt{\varpi \mu} \ll q_0 \ll \mu$, higher orders in the eft are needed.