cases, the system may be driven from a BCS-like state through a molecular Tonks-Girardeau gas close to resonance to a weakly interacting Bose gas of dimers.'
author:
- 'A. Recati$^{1,2}$, J.N. Fuchs$^{1,3}$, and W. Zwerger$^{1}$'
title: 'Boson-Fermion Resonance Model in One Dimension'
---

Introduction
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In this article, we consider the problem of attractive fermions in one dimension (1D), having in mind current experiments on ultra-cold two-component Fermi gases of atoms [@Regal; @crossoverexp]. In these systems, the $s$-wave interaction between fermions in different internal states can be tuned using a Feshbach resonance. By changing the interaction from weakly attractive to weakly repulsive via a resonance where the interaction diverges, one can explore the crossover from a BCS superfluid, when the attraction is weak and pairing only appears in momentum space, to a Bose-Einstein condensate (BEC) of molecular dimers [@crossoverth]. Experiments are currently investigating gases that are in a three dimensional regime (3D). A different situation occurs if the gas is confined in a very anisotropic cigar-shaped trap, like, e.g., in an atomic wire created with optical lattices [@Esslinger] or on an atom chip [@Reichel]. If the transverse confinement is strong enough, the system effectively becomes 1D, i.e., the radial degrees of freedom are frozen. We will refer to such a situation as quasi-1D. In this case, at zero temperature, a crossover takes place between a BCS-like state and a weakly interacting Bose gas of dimers. This crossover can be described by an exactly solvable model (the so-called modified Gaudin-Yang model) [@FRZ; @Tokatly], which is just a combination of the Gaudin-Yang model for attractive fermions [@GY] and of the Lieb-Liniger model for repulsive dimers [@LL]. Despite the fact, that there is no genuine off-diagonal long range order in 1D even at zero temperature, we refer to this situation as a one dimensional version of the BCS-BEC crossover.

Such a crossover can be realized in two ra