Effective Nonlinear Schr\"odinger Equations for Cigar-Shaped and Disk-Shaped Fermi Superfluids at Unitarity

TL;DR

This paper derives effective 1D and 2D nonlinear Schrödinger equations for superfluid Fermi gases at unitarity under tight confinement, simplifying the 3D problem while maintaining accuracy for different particle numbers.

Contribution

It introduces nonpolynomial Schrödinger equations for cigar- and disk-shaped superfluids, extending the reduction of the 3D theory to lower dimensions with validated results.

Findings

Effective 1D and 2D equations match 3D results.

Nonlinearity simplifies to power-law form for small and large N.

Equations are suitable for phenomenological applications.

Abstract

In the case of tight transverse confinement (cigar-shaped trap) the three-dimensional (3D) nonlinear Schr\"odinger equation, describing superfluid Fermi atoms at unitarity (infinite scattering length $|a|\to \infty$), is reduced to an effective one-dimensional form by averaging over the transverse coordinates. The resultant effective equation is a 1D nonpolynomial Schrodinger equation, which produces results in good agreement with the original 3D one. In the limit of small and large fermion number $N$ the nonlinearity is of simple power-law type. A similar reduction of the 3D theory to a two-dimensional form is also performed for a tight axial confinement (disk-shaped trap). The resultant effective 2D nonpolynomial equation also produces results in agreement with the original 3D equation and has simple power-law nonlinearity for small and large $N$. For both cigar- and disk-shaped superfluids our nonpolynomial Schr\"odinger equations are quite attractive for phenomenological application.