Coupled-pair approach for strongly interacting trapped fermionic atoms

We present a coupled-pair approach for studying few-body physics in harmonically trapped ultracold gases. The method is applied to a two-component Fermi system of $N$ particles. A stochastically variational Gaussian expansion method is applied, focusing on optimization of the two-body correlations present in the strongly interacting, or unitary, limit. The ground-state energy of the four-, six- and eight-body problem with equal spin populations is calculated with high accuracy and minimal computational effort. We also calculate the structural properties of these systems and discuss their implication for the many-body ultracold gas and other few-body calculations.

Figure 1

The three sets of linearly independent relative coordinates for $N=4$. (a) and (b) Conventional Jacobi vectors; (c) the fully paired $\mathsf{H}$-type channel. Each line represents the correlation between the centers of mass of two subsets of particles. All other channels can be obtained by applying the antisymmetrizer to one of these three channels so they contribute nothing extra to the solution.

Figure 2

The scaled pair correlation function $4\pi r^2{P}_{12}\left(r\right)/a_{\mathrm{ho}}^{-1}$ of the four-body ground state at unitarity using different combinations of channels. The calculation is performed for $r_0/a_{\mathrm{ho}}=0.01$ using only the $\mathsf{H}$-type channel (blue, solid), only the $\mathsf{K}$-type channels (red, dashed), and all three (black, dotted). The inset demonstrates the robustness of the calculation using only the $\mathsf{H}$-type channel. For each calculation the basis used was the same size for each channel so the $\mathsf{H}$-type-only calculation had the smallest basis.

Figure 3

(a) The $\mathsf{H}$-type channel for the $N=6$ problem; (b) and (c) the $\mathsf{H}$-type channels for the $N=8$ problem. All other fully paired channels can be obtained by applying the antisymmetrizer to one of these channels so they contribute nothing extra to the solution.

Figure 4

(a) Reduced one-body density $P_1\left(r\right)/a_{\mathrm{ho}}^{-3}$ at unitarity for $N=4$ (black, solid), $N=6$ (blue, dashed), and $N=8$ (red, dotted). In each case calculations were performed using the smallest possible $r_0$ and largest possible basis size. (b) Reduced (and scaled) pair correlation $4\pi r^2{P}_{12}\left(r\right)/a_{\mathrm{ho}}^{-1}$ for the same cases.