Interacting Fermions in Highly Elongated Harmonic Traps

Quasi-one-dimensional two-component Fermi gases with effectively attractive and repulsive interactions are characterized for arbitrary interaction strength. The ground-state properties of the gas confined in highly elongated harmonic traps are determined within the local density approximation. For strong attractive effective interactions the existence of a molecular Tonks-Girardeau gas is predicted. The frequency of the lowest breathing mode is calculated as a function of the coupling strength for both attractive and repulsive interactions.

Figure 1

Effective 1D coupling constant $g_{1\mathrm{D}}$ [solid line, Eq. (4)], together with effective 1D scattering length $a_{1\mathrm{D}}$ [dashed line, Eq. (5)] as a function of $a_{3\mathrm{D}}$.

Figure 2

$E_{\mathrm{h}\mathrm{o}\mathrm{m}}/N$ (solid line), ${\mu }_{\mathrm{h}\mathrm{o}\mathrm{m}}$ (dashed line), and $c$ (inset) for a homogeneous two-component 1D Fermi gas as a function of $\gamma$ (horizontal arrows indicate the asymptotic values of $E_{\mathrm{h}\mathrm{o}\mathrm{m}}/N$, ${\mu }_{\mathrm{h}\mathrm{o}\mathrm{m}}$, and $c$, respectively).

Figure 3

Energy per particle, $E/N-\hslash {\omega }_{\rho }$ (solid lines), and size of the cloud, $R$ (dashed lines), for an inhomogeneous two component 1D Fermi gas as a function of $N{a}_{1\mathrm{D}}^2/a_z^2$ for repulsive ($g_{1\mathrm{D}}>0$) and attractive ($g_{1\mathrm{D}}<0$) interactions.

Figure 4

Square of the lowest breathing mode frequency, ${\omega }^2$, as a function of the coupling strength $N{a}_{1\mathrm{D}}^2/a_z^2$ for an inhomogeneous two-component 1D Fermi gas with repulsive ($g_{1\mathrm{D}}>0$) and attractive ($g_{1\mathrm{D}}<0$) interactions determined numerically from Eq. (12) (solid lines). Dashed lines show analytic expansions (see text).