Multicomponent strongly attractive Fermi gas: A color superconductor in a one-dimensional harmonic trap

Recent advances in ultracold atomic Fermi gases make it possible to achieve a fermionic superfluid with multiple spin components. In this context, any mean-field description is expected to fail, owing to the presence of tightly bound clusters or molecules that consist of more than two particles. Here we present a detailed study of a strongly interacting multicomponent Fermi gas in a highly elongated or quasi-one-dimensional harmonic trap, which could be readily obtained in experiment. By using the exact Bethe ansatz solution and a local density approximation treatment of the harmonic trap, we investigate the equation of state of the multicomponent Fermi gas in both a homogeneous and trapped environment, as well as the density profiles and low-energy collective modes. The binding energy of multicomponent bound clusters is also given. We show that there is a peak in the collective mode frequency at the critical density for a deconfining transition to a many-body state that is analogous to the quark color superconductor state expected in neutron stars.

Figure 1

(Color online) One-dimensional interaction parameters for a three-component lithium gas in a two-dimensional optical lattice above the Feshbach resonances. Above a magnetic field $B=1000\mathrm{G}$, the difference of interaction parameters between different channels becomes very small, i.e., $({\delta }_{12}-{\delta }_{23})∕{\delta }_{12}<0.06$ and $({\delta }_{12}-{\delta }_{13})∕{\delta }_{12}<0.16$. As a result, a three-component lithium gas can be well described by our exactly soluble model of 1D Fermi gases with a symmetric interaction.

Figure 2

(Color online) Quasimomentum distributions (a) as a function of the dimensionless coupling constant or (b) as a function of the number of components. The distribution approaches $\kappa ∕2\pi$ for a large interaction strength. While in the weak coupling limit, it goes to $1∕\left(2\pi \right)$, but with a sharp increase at the boundary of $x=\pm 1$.

Figure 3

(Color online) Schematic diagram of interacting bound clusters for $\kappa =3$, in the low density limit.

Figure 4

(Color online) (a) Dependence of the uniform ground state energy per particle, (b) the chemical potential, and (c) the velocity of sound on the dimensionless coupling constant $\gamma$, at several number of species as indicated. The energy and sound velocity are in units of the Fermi energy ${\epsilon }_{F,\kappa }\equiv [{\hslash }^2{n}^2∕(2m\left)\right]({\pi }^2∕{\kappa }^2)$ and the Fermi velocity ${\nu }_{F,\kappa }\equiv [\hslash n∕m](\pi ∕\kappa )$, respectively. Thin solid lines are the analytic results in the two limiting cases for $\kappa =2$, as described in Eq. (3.20) and Eq. (3.29).

Figure 5

(Color online) Density profiles of a 1D trapped multicomponent Fermi cloud at three interaction parameters (a) $\delta =0.1$, (b) $\delta =1.0$, and (c) $\delta =10$. The linear density and the coordinate are in units of the peak density $n_{\mathrm{TF},\kappa }^0$ and Thomas-Fermi radius $x_{\mathrm{TF},\kappa }^0$ of an ideal gas, respectively.

Figure 6

(Color online) Thomas-Fermi radius, in units of $x_{\mathrm{TF},\kappa }^0$, as a function of the interaction parameter, at several number of component as indicated.

Figure 7

(Color online) Chemical potential as a function of the interaction parameter $\delta$. It is in units of $E_{F,\kappa }=N\hslash \omega ∕\kappa$. Thin solid lines are the analytic results in the two limiting cases for $\kappa =2$, as described in Eqs. (4.17, 4.21).

Figure 8

(Color online) Frequency corrections (with respect to the ideal gas value) of the lowest breathing modes $\delta {\omega }_B$ as a function of the dimensionless coupling parameter $\delta$, at several number of species as indicated.

Figure 9

(Color online) Interaction strength dependence of the frequency corrections $\delta {\omega }_n={\omega }_n-n\omega$ of the low-lying collective modes of a 1D three-component Fermi gas. Thin solid lines are the analytic sum-rule expansions for the weakly and strongly interacting limits, as described in Eqs. (4.29, 4.30), respectively.