Dilute Fermi Gas in Quasi-One-Dimensional Traps: From Weakly Interacting Fermions via Hard Core Bosons to a Weakly Interacting Bose Gas

We study equilibrium properties of a cold two-component Fermi gas confined in a quasi-one-dimensional trap of the transverse size l_⊥. In the dilute limit (nl_⊥1, where n is the 1D density) the problem is exactly solvable for an arbitrary 3D fermionic scattering length aF. When l_⊥/aF goes from -∞ to +∞, the system successively passes three regimes: weakly interacting Fermi gas, hard core Bose gas, and weakly coupled Bose gas. The regimes are separated by two crossovers at aF~±nl_⊥². In conclusion, we discuss experimental implications of these results.