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

TL;DR

This paper calculates gradient corrections to the local density approximation for a trapped superfluid Fermi gas near the Feshbach resonance, revealing their small magnitude and universal correction factors.

Contribution

It introduces a method to compute gradient corrections including Cooper pairing effects and derives a universal prefactor for the von Weizsäcker correction.

Findings

Gradient corrections are proportional to (d/R_TF)^4.

Universal correction factor κ_W=7/27 for von Weizsäcker term.

Corrections are small but systematically quantifiable.

Abstract

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 $(d/R_{TF})^4$, where $d$ is the oscillator length of the trap potential and $R_{TF}$ is the radial extension of the density $n$ in the Thomas-Fermi approximation. In particular at the Feshbach resonance the universal %constant $A_{TF}$ has the %correction in the center $A=A_{TF}+A_2(d/R_{TF})^4+\...$ corrections to the local density approximation are calculated and a universal prefactor $\kappa_W=7/27$ is derived for the von Weizs\"acker type correction $\kappa_W(\hbar^2/2m)(\nabla^2 n^{1/2}/n^{1/2})$.