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(perpendicular). In the dilute limit (nl(perpendicular)<1, where n is the 1D density) the problem is exactly solvable for an arbitrary 3D fermionic scattering length aF. When l(perpendicular)/aF goes from -infinity to +infinity, 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 approximately +/-nl2(perpendicular). In conclusion, we discuss experimental implications of these results.