Local density approximation for a perturbative equation of state

TL;DR

This paper develops a local density approximation method to predict properties of ultracold gases in harmonic traps based on a perturbative equation of state, enabling precise experimental measurements of quantum systems.

Contribution

It introduces a formalism that connects perturbative equations of state with observable properties of trapped quantum gases across various dimensions.

Findings

Predicted chemical potential, energies, and density profiles for trapped gases.

Calculated breathing mode frequencies for different geometries.

Applied the formalism to Bose, Fermi, and integrable 1D models.

Abstract

The knowledge of a series expansion of the equation of state provides a deep insight into the physical nature of a quantum system. Starting from a generic ``perturbative'' equation of state of a homogeneous ultracold gas we make predictions for the properties of the gas in the presence of harmonic confinement. The local density approximation is used to obtain the chemical potential, total and release energies, Thomas-Fermi size and density profile of a trapped system in three-, two-, and one- dimensional geometries. The frequencies of the lowest breathing modes are calculated using scaling and sum-rule approaches and could be used in an experiment as a high precision tool for obtaining the expansion terms of the equation of state. The derived formalism is applied to dilute Bose and Fermi gases in different dimensions and to integrable one-dimensional models. Physical meaning of expansion terms in a number of systems is discussed.