Gradient corrections to the local-density approximation for trapped superfluid Fermi gases

Two species superfluid Fermi gas is investigated on the BCS side up to the Feshbach resonance. Using the Greens’s function technique gradient corrections are calculated to the generalized Thomas-Fermi theory including Cooper pairing. Their relative magnitude is found to be measured by the small parameter $(d/R_{\mathrm{TF}}){}^4$, where $d$ is the oscillator length of the trap potential and $R_{\mathrm{TF}}$ is the radial extension of the density $n$ in the Thomas-Fermi approximation. In particular, at the Feshbach resonance the universal corrections to the local density approximation are calculated and a universal prefactor ${\kappa }_W=7/27$ is derived for the von Weizsäcker–type correction ${\kappa }_W(\hslash {}^2/2m)({\nabla }^2{n}^{1/2}/n^{1/2})$.

Figure 1

The dimensionless function $j_1(t)$ defined in Eq. (A4).

Figure 2

The dimensionless function $i_1(t)$ defined in Eq. (A3).

Figure 3

Solutions of the gap equation without (solid line) and with (dashed line) the second-order gradient corrections as a function of the dimensionless radius $\rho =R/R_{\mathrm{TF}}$, and measured in units of $\hslash {\omega }_0=h\nu$ (see text). Parameters are $a/d=1/(3\pi )$, $R_{\mathrm{TF}}/d=8$.

Figure 4

Density profiles without (solid line) and with (dashed line) the second-order gradient corrections as a function of the dimensionless radius $\rho =R/R_{\mathrm{TF}}$, and measured in units of $1/d^3$ Parameters are $a/d=1/(3\pi )$, $R_{\mathrm{TF}}/d=8$.

Figure 5

The $P_1(t)$ function [see Eq. (B3)].

Figure 6

The $Q_1(t)$ function [see Eq. (B4)].

Figure 7

The $P_2(t)$ function [see Eq. (B5)].

Figure 8

The $Q_2(t)$ function [see Eq. (B6)].