One-dimensional Fermi gas with a single impurity in a harmonic trap: Perturbative description of the upper branch

The transition from “few to many” has recently been probed experimentally in an ultracold harmonically confined one-dimensional lithium gas, in which a single impurity atom interacts with a background gas consisting of one, two, or more identical fermions [A. N. Wenz et al., Science 342, 457 (2013)]. For repulsive interactions between the background or majority atoms and the impurity, the interaction energy for relatively moderate system sizes was analyzed and found to converge toward the corresponding expression for an infinitely large Fermi gas. Motivated by these experimental results, we apply perturbative techniques to determine the interaction energy for weak and strong coupling strengths and derive approximate descriptions for the interaction energy for repulsive interactions with varying strength between the impurity and the majority atoms and any number of majority atoms.

Figure 1

(a) Solid, dotted, and dashed lines show the energy of the upper branch for $N=1$, 2, and 3 as a function of $-1/g$. The energies of the $(1,1)$ system are obtained by solving the transcendental equation derived in Ref. [21]. The energies of the $(2,1)$ and $(3,1)$ system are taken from Refs. [14, 15, 22]. (b) Solid, dotted, and dashed lines show the interaction energy $\epsilon \left(N\right)$, normalized by the Fermi energy $E_{\text{F}}\left(N\right)$, for $N=1$, 2, and 3, respectively, as a function of $-1/g$. The harmonic oscillator length $a_{\text{ho}}$ is defined in Eq. (10).

Figure 2

Symbols show the coefficients (a) $C^{\left(1\right)}\left(N\right)$, (b) $C^{\left(2\right)}\left(N\right)$, and (c) $C^{\left(3\right)}\left(N\right)$ as a function of $1/N$. The solid lines show our fits with $j_{\text{max}}=6$, 3, and 3, respectively.

Figure 3

Solid lines show the quantity $\rho \left(N\right)$ as a function of $-1/\gamma$ for (a) $N=2$, (b) $N=3$, and (c) $N=\infty$, respectively. For comparison, dotted, dashed, and dash-dotted lines show the perturbative results for $\rho \left(N\right)$ accounting for terms up to order ${\gamma }^0,{\gamma }^1$, and ${\gamma }^2$, respectively, in the weakly interacting regime and accounting for terms up to order ${\gamma }^{-1},{\gamma }^{-2}$, and ${\gamma }^{-3}$, respectively, in the strongly interacting regime.

Figure 4

The quantity $\delta (N,N^{\prime })$ as a function of $-1/\gamma$. The solid line is for (a) $N=2$ and $N^{\prime }=\infty$ and the circles are for (b) $N=3$ and $N^{\prime }=\infty$ and (c) $N=2$ and $N^{\prime }=3$. In the large $\gamma$ regime, the uncertainty of the numerically determined $(3,1)$ energies leads to appreciable uncertainties in $\delta (2,3)$ and $\delta (3,\infty )$ [see the error bars in Figs. 4 and 4]. For comparison, dotted, dashed, and dash-dotted lines show the perturbative results for $\delta (N,N^{\prime })$ accounting for terms up to order ${\gamma }^0,{\gamma }^1$, and ${\gamma }^2$, respectively, in the weakly interacting, small $\left|\gamma \right|$ regime and accounting for terms up to order ${\gamma }^{-1},{\gamma }^{-2}$, and ${\gamma }^{-3}$, respectively, in the strongly interacting, large $\left|\gamma \right|$ regime.