Thomas-Fermi von Weizsacker theory for a harmonically trapped, two-dimensional, spin-polarized dipolar Fermi gas

We systematically develop a density functional description for the equilibrium properties of a two-dimensional, harmonically trapped, spin-polarized dipolar Fermi gas based on the Thomas- Fermi von Weizsacker approximation. We pay particular attention to the construction of the two- dimensional kinetic energy functional, where corrections beyond the local density approximation must be motivated with care. We also present an intuitive derivation of the interaction energy functional associated with the dipolar interactions, and provide physical insight into why it can be represented as a local functional. Finally, a simple, and highly efficient self-consistent numerical procedure is developed to determine the equilibrium density of the system for a range of dipole interaction strengths. The ability to fabricate such systems in the laboratory now opens the door for the investigation of both the equilibrium and dynamical properties of dipolar Fermi gases, and will enable contact to be made with the large body of theoretical work already in the literature (1). Moreover, it is now reasonable to expect that quasi-2D degenerate dipolar Fermi gases will also be realized experimentally, thereby allowing for studies into the stability, pairing, and superfluidity of low-dimensional dipolar systems, which to date have only been investigated theoretically (2, 8, 9). With a view towards ultimately calculating the collective mode frequencies, we develop in this paper a density- functional theory (DFT) for the equilibrium properties of a degenerate, harmonically trapped, spin polarized dipolar Fermi gas. Our theoretical framework is based on the Thomas-Fermi von Weizsacker (TFvW) approximation which was previously formulated in the context of degenerate electron gases (10). The mathematical framework of the TFvW theory is very simple, numerically easy to implement, and computationally inexpensive. The TFvW theory has also been shown to provide an exceedingly accurate description of equilibrium properties, as well as collective excitations (i.e., magnetoplasmons), of electronic systems in a variety of two-and three-dimensional confinement geometries (10- 16). Our purpose here is to take advantage of this approach, which is largely unknown in the cold-atoms community, and apply it to the dipolar Fermi gas. We will only address the equilibrium properties in this paper, and leave the presentation of the more involved mode calculation to a future publication. Moreover, in anticipation of forthcoming experiments, along with the goal of making contact with the recent theoretical work of Fang and Englert (17), we will