Molecular branch of a small highly-elongated Fermi gas with an impurity: Full three-dimensional versus effective one-dimensional description

We consider an impurity immersed in a small Fermi gas under highly-elongated harmonic confinement. The impurity interacts with the atoms of the Fermi gas through an isotropic short-range potential with three-dimensional free-space s-wave scattering length $a_{3\text{d}}$. We investigate the energies of the molecular branch, i.e., the energies of the state that corresponds to a gas consisting of a weakly-bound diatomic molecule and "unpaired" atoms, as a function of the s-wave scattering length $a_{3\text{d}}$ and the ratio $\eta$ between the angular trapping frequencies in the tight and weak confinement directions. The energies obtained from our three-dimensional description that accounts for the dynamics in the weak and tight confinement directions are compared with those obtained within an effective one-dimensional framework, which accounts for the dynamics in the tight confinement direction via a renormalized one-dimensional coupling constant. Our theoretical results are related to recent experimental measurements.