Trapped system

Using the solutions for the homogeneous two-component 1D Fermi gas,

Figure 8.3: Energy per particle, $E/N-\hbar \omega _\rho$ (solid lines), and size of the cloud, $R$ (dashed lines), for an inhomogeneous two component 1D Fermi gas as a function of $Na_{1D}^2/a_z^2$ for repulsive ($g_{1D}>0$) and attractive ($g_{1D}<0$) interactions.

we now describe the inhomogeneous gas, Eq. 8.2, within the LDA [DLO01,MS02,RFZZ03a]. This approximation is applicable if the size $R$ of the cloud is much larger than the harmonic oscillator length $a_z$ in the longitudinal direction, $a_z=\sqrt{\hbar/m\omega_z}$, implying $\epsilon_F\gg \hbar\omega_z$ and $N\gg 1$. The chemical potential $\mu$ of the inhomogeneous system can be determined from the local equilibrium condition,

$$\mu=\mu_{hom}[n_{1D}(z)]+\frac{1}{2}m\omega_z^2z^2,$$ (8.10)

and the normalization condition $N=\int_{-R}^R n_{1D}(z)dz$, where $z$ is measured from the center of the trap, $R=\sqrt{2\mu^{\prime}/(m\omega_z^2)}$, and $\mu^{\prime}=\mu$ for $g_{1D}>0$ and $\mu^{\prime}=\mu+\vert\epsilon_{bound}\vert/2$ for $g_{1D}<0$. The normalization condition can be reexpressed in terms of the dimensionless chemical potential $\tilde{\mu}$ and the dimensionless density $\tilde{n}_{1D}$ [ $\tilde{\mu}=\mu^{\prime}/(\hbar^2/2ma_{1D}^2)$ and $\tilde{n}_{1D}=\vert a_{1D}\vert n_{1D}$],

$$N\frac{a_{1D}^2}{a_z^2}= \int_0^{\tilde{\mu}} \frac{\tilde{n}_{1D}(\tilde{\mu}-x)}{\sqrt{x}} dx\;.$$ (8.11)

This expression emphasizes that the coupling strength is determined by $Na_{1D}^2/a_z^2$; $Na_{1D}^2/a_z^2\gg 1$ corresponds to the weak coupling and $Na_{1D}^2/a_z^2\ll 1$ to the strong coupling regime, irrespective of whether the interactions are attractive or repulsive [MS02].

Figure 8.3 shows the energy per particle $E/N$ and the size $R$ of the cloud as a function of the coupling strength $Na_{1D}^2/a_z^2$ for positive and negative $g_{1D}$ calculated within the LDA for an inhomogeneous two-component 1D Fermi gas. Compared to the non-interacting gas, for which $R=\sqrt{N}a_z$, $R$ increases for repulsive interactions and decreases for attractive interactions. For $Na_{1D}^2/a_z^2\ll 1$, $R$ reaches the asymptotic value $\sqrt{2N}a_z$ for the strongly repulsive regime, $g_{1D} \rightarrow +\infty$, and the value $\sqrt{N/2}a_z$ for the strongly attractive regime, $g_{1D} \rightarrow -\infty$. The shrinking of the cloud for attractive interactions reflects the formation of tightly bound molecules. In the limit $g_{1D} \rightarrow -\infty$, the energy per particle approaches $\epsilon_{bound}/2+N \hbar \omega_z/8 + \hbar \omega_{\rho}$, indicating the formation of a molecular bosonic Tonks-Girardeau gas, consisting of $N/2$ molecules. The size of the cloud shrinks from $R=\sqrt{2N}a_z$ in the strongly repulsive regime ( $Na_{1D}^2/a_z^2\ll 1$ and $g_{1D}>0$) to $R=\sqrt{N/2}a_z$ in the strongly attractive regime ( $Na_{1D}^2/a_z^2\ll 1$ and $g_{1D}<0$). We also notice that, similarly to the homogeneous case, for large attractive interactions the energy per particle approaches the molecular binding energy $\epsilon_{bound}$.

Using a sum rule approach, the frequency $\omega$ of the lowest compressional (breathing) mode of harmonically trapped 1D gases can be calculated from the mean-square size of the cloud $\langle z^2 \rangle$ [MS02],

$$\omega^2=-2\frac{\langle z^2\rangle }{d\langle z^2\rangle/d\omega_z^2}$$ (8.12)

In the weak and strong coupling regime ( $Na_{1D}^2/a_z^2\gg 1$ and $\ll 1$, respectively), $\langle z^2 \rangle$ has the same dependence on $\omega_z$ as the ideal 1D Fermi gas. Consequently, $\omega$ is in these limits given by $2\omega_z$, irrespective of whether the interaction is repulsive or attractive. Solid lines in Fig. 8.3 show $\omega ^2$, determined numerically from Eq. 8.12, as a function of the interaction strength $Na_{1D}^2/a_z^2$. A non-trivial behavior of $\omega ^2$ as a function of $Na_{1D}^2/a_z^2$ is visible. To gain further insight, we calculate the first correction $\delta\omega$ to the breathing mode frequency $\omega$

Figure 8.4: Square of the lowest breathing mode frequency, $\omega ^2$, as a function of the coupling strength $Na_{1D}^2/a_z^2$ for an inhomogeneous two-component 1D Fermi gas with repulsive ($g_{1D}>0$) and attractive ($g_{1D}<0$) interactions determined numerically from Eq. 8.12 (solid lines). Dashed lines show analytic expansions.

[ $\omega=2\omega_z(1+\delta\omega/\omega_z+\cdots)$] analytically for weak repulsive and attractive interactions, as well as for strong repulsive and attractive interactions. For the weak coupling regime, we find $\delta\omega/\omega_z=\pm (4/3\pi^2)/(Na_{1D}^2/a_z^2)^{1/2}$, where the minus sign applies to repulsive interactions and the plus sign to attractive interactions. For the strong coupling regime, we find $\delta\omega/\omega_z=-[16\sqrt{2}\ln(2)/15\pi^2](Na_{1D}^2/a_z^2)^{1/2}$ for repulsive interactions and $\delta\omega/\omega_z=(8\sqrt{2}/15\pi^2)(Na_{1D}^2/a_z^2)^{1/2}$ for attractive interactions (see Table 1.1).Dashed lines in Fig. 8.4 show the resulting analytic expansions for $\omega ^2$, which describe the lowest breathing mode frequency quite well over a fairly large range of interaction strengths but break down for $Na_{1D}^2/a_z^2\sim 1$.