Model

Consider a two-component atomic Fermi gas confined in a highly-elongated trap. The fermionic atoms are assumed to belong to the same atomic species, that is, to have the same mass $m$, but to be trapped in different hyperfine states $\sigma$, where $\sigma$ represents a generalized spin or angular momentum, $\sigma=\uparrow$ or $\downarrow$. The trapping potential is assumed to be harmonic and axially symmetric,

$$V_{trap}=\sum_{i=1}^N\frac{1}{2}m\left( \omega_\rho^2\rho_i^2 + \omega_z^2 z_i^2 \right)$$ (8.1)

Here, $\rho_i=\sqrt{x_i^2+y_i^2}$ and $z_i$ denote, respectively, the radial and longitudinal coordinate of the $i$th atom; $\omega_{\rho}$ and $\omega_z$ denote, respectively, the angular frequency in the radial and longitudinal direction; and $N$ denotes the total number of atoms. We require the anisotropy parameter $\lambda$, $\lambda=\omega_z/\omega_\rho$, to be so small that the transverse motion is ``frozen'' to zero point oscillations. At zero temperature this implies that the Fermi energy associated with the longitudinal motion of the atoms in the absence of interactions, $\epsilon_F=N\hbar\omega_z/2$, is much smaller than the separation between the levels in the transverse direction, $\epsilon_F\ll\hbar\omega_\rho$. This condition is fulfilled if $\lambda\ll 1/N$. The outlined scenario can be realized experimentally with present-day technology using optical traps.

If the Fermi gas is kinematically in 1D, it can be described by an effective 1D Hamiltonian with contact interactions,

$$H=N\hbar\omega_\rho + H_{1D}^0 + \sum_{i=1}^N\frac{1}{2}m\omega_z^2z_i^2,$$ (8.2)

where

$$H_{1D}^0=-\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\p... ...um_{i=1}^{N_\uparrow} \sum_{j=1}^{N_\downarrow}\delta(z_i-z_j)$$ (8.3)

and $N=N_{\uparrow}+N_{\downarrow}$. This effective Hamiltonian accounts for the interspecies atom-atom interactions, which are parameterized by the 3d $s$-wave scattering length $a_{3d}$, through the effective 1D coupling constant $g_{1D}$ [Ols98,BMO03],

$$g_{1D}=\frac{2\hbar^2a_{3d}}{m a_\rho^2} \frac{1}{1-A a_{3d}/a_\rho},$$ (8.4)

but neglects the typically much weaker $p$-wave interactions. In Eq. 8.4, $a_\rho=\sqrt{\hbar/m\omega_\rho}$ is the characteristic oscillator length in the transverse direction and $A=\vert\zeta(1/2)\vert/\sqrt{2}\simeq 1.0326$. Alternatively, $g_{1D}$ can be expressed through the effective 1D scattering length $a_{1D}$, $g_{1D} = - 2\hbar^2/(ma_{1D})$, where

$$a_{1D}=-a_\rho\left(\frac{a_\rho}{a_{3d}}-A\right)$$ (8.5)

Figure 8.1 shows $g_{1D}$ and $a_{1D}$ as a

Figure: Effective 1D coupling constant $g_{1D}$ [solid line, Eq. 8.4], together with effective 1D scattering length $a_{1D}$ [dashed line, Eq. 8.5] as a function of $a_{3d}$.

function of the 3d $s$-wave scattering length $a_{3d}$, which can be varied continuously by application of an external field. The effective 1D interaction is repulsive, $g_{1D}>0$, for $0<a_{3d}<a_{3d}^c$ ( $a_{3d}^c=0.9684 a_\rho$), and attractive, $g_{1D}<0$, for $a_{3d}>a_{3d}^c$ and for $a_{3d}<0$. By varying $a_{3d}$, it is possible to go adiabatically from the weakly-interacting regime ($g_{1D}\sim 0$) to the strongly-interacting repulsive regime ( $g_{1D}\to +\infty$ or $a_{3d}\lesssim a_{3d}^c$), as well as from the weakly-interacting regime to the strongly-interacting attractive regime ( $g_{1D}\to-\infty$ or $a_{3d}\gtrsim a_{3d}^c$)^8.2

For two fermions with different spin the Hamiltonian $H_{1D}^0$, Eq. 8.3, supports one bound state with binding energy $\epsilon_{bound}=-\hbar^2/(ma_{1D}^2)$ and spatial extent $\sim a_{1D}$ for $g_{1D}<0$, and no bound state for $g_{1D}>0$, that is, the molecular state becomes exceedingly weakly-bound and spatially-delocalized as $g_{1D}\to 0^-$ [Ols98,BMO03]. In the following we investigate the properties of a gas with $N$ fermions, $N_{\uparrow}= N_{\downarrow}$, for both effectively attractive and repulsive 1D interactions with and without longitudinal confinement.