Supershell structures and pairing in ultracold trapped Fermi gases

We calculate level densities and pairing gaps for an ultracold dilute gas of fermionic atoms in harmonic traps under the influence of mean field and anharmonic quartic trap potentials. Supershell nodes, which were found in Hartree-Fock calculations, are calculated analytically within periodic orbit theory as well as from WKB calculations. For attractive interactions, the underlying level densities are crucial for pairing and supershell structures in gaps are predicted.

Figure 1

(Color online) The upper figure (a) shows the leading term $(k=1)$ of the oscillating part of the perturbative level density of Eq. (9) as a function of $n_F=E∕\hslash \omega$, for the case of an external potential $V=V_{\mathrm{HO}}+\epsilon r^4$ with $\epsilon =2∕{40}^2\approx 0.0013$. The middle and lower figures [(b) and (c)] show the oscillating part of the total energy according to a numerical HF calculation [12], with interaction strength $2\pi a=1$ and $2\pi a=0.1$, as a function of the HO shell number ($\hslash =\omega =1$ in all figures). This illustrates qualitatively that a supernode, e.g., at $n_F=40$, can be due to interaction (b) and/or an additional quartic term to the HO potential (a).

Figure 2

(Color online) Dimensionless multishell pairing gaps for a HO trap with an additional quartic term in the potential with $\epsilon =2∕{40}^2$, i.e., for the level density of Fig. 1a with supernodes at $n_F\simeq 40$, $40\sqrt{2}\approx 57$, etc. The interaction strength is $a=-0.05$ [top (red) curve], $a=-0.03$ [middle (blue) curve, with the inset figure around the first supernode], and $a=-0.01$ [lower (green) curve]. In the inset plot it is clearly seen that the local minima for $l\sim n_F$ and $l\sim 0$ before the supernode turns into local maxima after the supernode, as a consequence of overlapping shells. The dashed (red) line is the multishell gap $\Delta =G∕[1-2G\ln (n_F)∕\hslash \omega ]$ for $a=-0.05$ and the upper and/or lower thin solid lines (black) are the single middle- and/or end-shell pairing for $a=-0.01$ (see text).