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

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.

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