Density Profile of a Harmonically Trapped Ideal Fermi Gas in Arbitrary Dimension

Abstract

Closed form analytic expressions are derived for the density profile of a harmonically trapped noninteracting Fermi gas in d dimensions. Shell structure effects are included to leading order in 1/N , where N is the number of particles. These corrections to the local density approximation scale as δn/n ∼ N −α , where α = (1 + 1/d)/2.PACS numbers: 03.75.SsExperiments on quantum degenerate Fermi atoms have motivated a series of theoretical studies of the basic properties of zero [1,2,3,4,5,6] and finite [7] temperature noninteracting fermions in inhomogeneous potentials. Here we derive closed form analytic expressions for the ground state density of a harmonically trapped Fermi gas in d dimensions.Theoretical studies of noninteracting particles are crucial for understanding experiments on nonresonant Fermi gases [8]. A typical length scale for interactions is r 0 ≈nm, and a typical interparticle spacing is n −1/3 ≈ µm≫ r 0 . Thus, even in multiple component gases, where s-wave collisions are allowed, the interaction energy per particle E int ≈h 2 r 0 n/m is small compared with the Fermi energy E f ≈h 2 n 2/3 /m. Therefore the interactions can be ignored for calculating gross features of the ground state structure. This separation of energy scales is even more dramatic in a spin-polarized gas, where swave collisions are forbidden, and therefore p-wave collisions dominate. The cross-section for p-wave collisions is down by a factor of (k f r 0 ) 4 relative to s-wave. Although our calculations are for arbitrary dimension, d, a particularly important case is d = 1 where theoretical predictions about noninteracting fermions can be applied to hard-core bosons [9].Most experiments are performed in harmonic traps, so it is natural to consider the density profile of a gas of N fermions in a potential V (r) = mω 2 r 2 /2. Since we ignore interactions, each spin component is independent, and it suffices to consider the spinless (or spin-polarized) case. The simplest expression for this density profile comes from the local density (Thomas-Fermi) approximation, where the trap potential gives a spatially dependent chemical potential µ(r) = k 2 f (r)/2m = µ 0 − V (r), and the local density is derived from the relationship between density and chemical potential in a d-dimensional homogeneous system, n = (k f /2 √ π) d /Γ(d/2 + 1), resulting in a density profile,whereis the radius of the cloud,and ℓ 2 =h 2 /mω 2 is the oscillator length. Although this is an excellent approximation to the true ground-state density profile, it misses "shell effects" resulting from the statistical correlations in an ideal Fermi gas. These shell effects give rise to corregations on top of this smooth profile which are analogous to the Friedel oscillations which occur in the density of a uniform Fermi gas near an impurity [10]. As we show below, these corregations scale as δn/n ∼ N −α where α = (1 + 1/d)/2. These deviations are therefore only significant for small numbers of particles. Despite their small size, previous estimates [1,11] suggest that the...

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