A simple variational approach for an interacting Fermi trapped gas

TL;DR

This paper introduces a variational method to analyze the quantum states of a two-component Fermi trapped gas, deriving explicit expressions for interaction energies applicable to both small and large particle numbers, highlighting the effects of interactions and temperature.

Contribution

It presents a novel variational approach with closed-form expressions for interaction energies in Fermi gases, valid across different particle counts and incorporating pairing and shell effects.

Findings

Interaction energy scales polynomially with Fermi energy at zero temperature.

The approach captures the discrete nature of energy levels and shell effects.

At large N, the interaction energy aligns with mean field approximation results.

Abstract

Quantum states of a two-component Fermi trapped gas are described by introducing an effective trap frequency, determined via variational techniques. Closed expressions for the contribution of a contact interaction potential to the total energy and the pairing interaction are derived. They are valid for both few and large number of particles, given the discrete nature of the formulation, and therefore richer than the continuous expressions, which are perfectly matched. Pairing energies within a shell are explicitly evaluated and its allowed values at a given energy level delimited. We show the importance of the interaction over the trap energy as the number of particles ($N$) grows and the temperature decreases. At zero temperature we find a polynomial dependence of the interaction energy on the Fermi energy, whose dominant term at large $N$ corresponds with the mean field approximation result. In addition, the role of the strength of an attractive potential on the total energy is exhibited.