Introduction

The study of cold quasi one dimensional atomic quantum gases presents a very active area of research. So far, most of the experimental [GVL+01,SKC+01,GBM+01,TOH+04,MSKE03] and theoretical [Ols98,PSW00,DLO01,MS02,GWT01] investigations have been devoted to quasi-one dimensional Bose gases and, in particular, to the strongly-interacting Tonks-Girardeau gas, which can be mapped to a gas of non-interacting fermions [Gir60,Ols98,RT03]. Quasi-1D two-component atomic Fermi gases have not been realized experimentally yet; however, their realization in highly-elongated, needle-shaped traps is within reach of present-day techniques. The behavior of quasi one dimensional two-component Fermi gases can, if the confinement is chosen properly, be characterized to a very good approximation by an effective 1D coupling constant, $g_{1D}$, which encapsulates the interspecies atom-atom interaction strength. This coupling constant can be tuned to essentially any value, including zero and $\pm\infty$, by varying the 3d $s$-wave scattering length $a_{3d}$ through application of an external magnetic field in the proximity of a Feshbach resonance.

The role of interactions in quasi one dimensional atomic Fermi gases has been studied mainly in connection with Luttinger liquid theory [XW02,GW04,RFZZ03a,RFZZ03b]. Recati et al. [RFZZ03a,RFZZ03b] investigate the properties of a two-component Fermi gas with repulsive interspecies interactions confined in highly-elongated harmonic traps. In the limit of weak and strong coupling these authors relate the parameters of the Luttinger Hamiltonian, which describe the low-energy properties of the gas, to the microscopic parameters of the system. The prospect of realizing Luttinger liquids with cold fermionic atoms is fascinating since it would allow detailed investigations of strongly correlated many-body systems, which play a central role in condensed matter physics^8.1, to be conducted.

In homogeneous 1D Fermi gases with attractive interactions, sound waves propagate with a well defined velocity, while spin waves exhibit a gap [KO75]. Furthermore, in the strong-coupling regime, the ground state is comprised of bosonic molecules (consisting of two fermions with different spin), whose spatial size is much smaller than the average intermolecular distance [KO75]. Consequently, BCS-type equations have been discussed for effectively attractive 1D interactions [CEE+91]. The quasi one dimensional molecular Bose gas discussed here (see also Ref. [Tok04,FRZ04]) has similarities with the formation of a molecular Bose-Einstein condensate (BEC) from a 3d Fermi sea close to a magnetic atom-atom Feshbach resonance [GRJ03,ZSS+03].

This Chapter investigates the properties of inhomogeneous quasi one dimensional two-component Fermi gases under harmonic confinement with attractive and repulsive interspecies interactions. Our study is based on the exact equation of state of a homogeneous 1D system of fermions with zero-range attractive [Gau67,KO75] and repulsive [Yan67] interactions treated within the local density approximation (Sec. 1.6). We calculate the energy per particle, the size of the cloud, and the frequency of the lowest compressional mode as a function of the effective 1D coupling constant, including infinitely strong attractive and repulsive interactions. Our predictions for the size of the cloud and for the breathing mode frequency have immediate implications for experimental studies. It has been shown recently for quasi one dimensional Bose gases [MSKE03] that precise measurements of collective mode frequencies can provide evidence for beyond mean-field effects. For attractive interactions we discuss the cross-over from the weak- to the strong-coupling regime and point out the possibility of forming a mechanically stable molecular Tonks-Girardeau gas.