Dimensional crossover in a Fermi gas from one to effective six dimensions and a cross-dimensional Tomonaga-Luttinger model

TL;DR

This paper investigates the dimensional crossover in a noninteracting Fermi gas within anisotropic traps, revealing how the dynamical structure factor mimics higher-dimensional systems and proposing a generalized Tomonaga-Luttinger model for multimode configurations.

Contribution

It introduces a method to simulate higher-dimensional Fermi gases using lower-dimensional traps and develops a generalized Tomonaga-Luttinger model for multimode systems.

Findings

Dynamical structure factor in a $d$-dimensional trap mimics an effective $2d$-dimensional box trap.

Proposes conditions to observe $d$-dimensional behavior in partial harmonic confinement.

The generalized Tomonaga-Luttinger model accurately describes low-energy backscattering in the 2D limit.

Abstract

We describe the dimensional crossover in a noninteracting Fermi gas in an anisotropic trap, obtained by populating various transverse modes of the trap. We study the dynamical structure factor and drag force. Starting from a dimension $d$, the $(\!d\!+\!1\!)$-dimensional case is obtained to a good approximation with relatively few modes. We show that the dynamical structure factor of a gas in a $d$-dimensional harmonic trap simulates an effective $2d$-dimensional box trap. We focus then on the experimentally relevant situation when only a portion of the gas in harmonic confinement is probed and give a condition to obtain the behavior of a $d$-dimensional gas in a box. Finally, we propose a generalized Tomonaga-Luttinger model for the multimode configuration and compare the dynamical structure factor in the 2D limit with the exact result, finding that it is accurate in the backscattering region and at low energy.