One- and two-dimensional reductions of the mean-field description of degenerate Fermi gases

TL;DR

This paper derives effective one- and two-dimensional equations for degenerate Fermi gases in various trapping potentials, including optical lattices, and validates these models through numerical simulations and analytical solutions.

Contribution

It introduces a variational approach to reduce 3D mean-field equations to effective 1D and 2D forms for Fermi gases, including analytical solutions for non-interacting cases.

Findings

Effective low-dimensional equations closely match 3D results.

Analytical solutions obtained for non-interacting 2D Fermi gases.

Density patterns depend on trap and lattice strengths, and scattering length.

Abstract

We study collective behavior of Fermi gases trapped in various external potentials, including optical lattices (OLs), in the framework of the mean-field (hydrodynamic) description. Using the variational method, we derive effective dynamical equations for the one- and two-dimensional (1D and 2D) settings from the general 3D mean-field equation. The respective confinement is provided by trapping potentials with the cylindrical and planar symmetry, respectively. The resulting equations are nonpolynomial Schr% \"{o}dinger equations (NPSEs) coupled to equations for the local transverse size of the trapped states. Numerical simulations demonstrate close agreement of results produced by the underlying 3D equation and the effective low-dimensional ones. We consider the ground state in these settings. In particular, analytical solutions are obtained for the effectively 2D non-interacting Fermi gas. Differences between the 1D and 2D configurations are highlighted. Finally, we analyze the dependence of the 1D and 2D density patterns of the trapped gas, in the presence of the OL, on the strengths of the confining and OL potentials, and on the scattering length which determines the strength of interactions between non-identical fermions.