Corrections to Local Density Approximation for superfluid trapped fermionic atoms from the Wigner-Kirkwood $\hslash$ expansion

Abstract

A semiclassical second-order differential equation for the inhomogeneous local gap $\Delta (r)$ is derived from a strict second-order $\hslash$ expansion of the anomalous pairing tensor and compared with a similar equation given by Simonucci et al. The second-order normal density matrix is given as well. Several extra gradient terms are revealed. Second-order expressions at finite temperature are given for the first time. The corresponding Ginzburg-Landau equation is presented and it is shown that, compared to the equation of Baranov and Petrov, an extra second-order gradient term is present. Applications to the pairing gap in cold atoms in a harmonic trap are presented.

Comments: 12 pages, 4 figures. To be submitted to the EPJ A Topical Collection devoted to the legacy of Peter Schuck