Crossover between few and many fermions in a harmonic trap

The properties of a balanced two-component Fermi gas in a one-dimensional harmonic trap are studied by means of the coupled-cluster method. For few fermions we recover the results of exact diagonalization, yet with this method we are able to study much larger systems. We compute the energy, the chemical potential, the pairing gap, and the density profile of the trapped clouds, smoothly mapping the crossover between the few-body and many-body limits. The energy is found to converge surprisingly rapidly to the many-body result for every value of the interaction strength. Many more particles are instead needed to give rise to the nonanalytic behavior of the pairing gap, and to smoothen the pronounced even-odd oscillations of the chemical potential induced by the shell structure of the trap.

Figure 1

Rescaled energies ${\mathcal{E}}_N=E_N/E_N^{\left(0\right)}$ for various number of fermions, as a function of the interaction strength $\gamma$. The squares and circles are, respectively, the results of the FCI and CC calculations. The black dashed line is the analytic two-body result ${\mathcal{E}}_2$ for $1+1$ particles [57], and the red dotted line is the thermodynamic result ${\mathcal{E}}_{\infty }$ obtained from the GY+LDA approach [18]. The inset shows the difference ${\mathcal{E}}_{\infty }-{\mathcal{E}}_2$ vs $\gamma$, which remains surprisingly small for every interaction strength.

Figure 2

Difference between the rescaled few-body energies and the analytical two-body result; solid circles (squares) are the CC (FCI) results, while the red dotted line is the GY+LDA result. The insets are vertical cuts across the main figure (i.e., at fixed $\gamma$), plotted as a function of $1/N$. The bright green dots are the expected many-body limit, and the open circles are the first-order perturbation approximation, Eq. (4).

Figure 3

BCS pairing gap ${\mathrm{\Delta }}_N$ of a few-fermion system with $N$ particles. Solid circles indicate CC results for increasing $N$, and the red squares are their extrapolation to $N\to \infty$. The red dotted line is the thermodynamic result, Eq. (5). Inset: Chemical potential ${\mu }_N$ (in units of $N\hslash \omega$) vs $1/N$, evaluated at $\gamma =-\pi /2$.

Figure 4

Density of a balanced two-component Fermi gas. The solid lines in the left panel display FCI results for $3+3$ particles, while the ones in the right panel show CC results for $15+15$ atoms. The dashed lines are the GY+LDA results, and the black dotted lines are analytic results for $\gamma \ll -1$ (LL), $\gamma =0$ (0), and $\gamma \gg 1$ (TG).