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

Cold-atom experiments that 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 effective-field theory (EFT) description of the unitary Fermi gas which, among other things, 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 Wentzel--Kramers--Brillouin 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. This work presents a systematic and quantitative method for treating the presence of a trapping potential, which is essential for interpreting experimental realizations of the unitary Fermi gas. It provides clear predictions for how the trapping potential modifies the dispersion relation's curvature, a key factor in characterizing the relaxation mechanisms of the superfluid. The most significant deviations from linear dispersion due to the trapping potential are found in the far-infrared region of the superfluid EFT.