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arXiv:cond-mat/0211102 · Physical Review A

Shell structure in the density profiles for noninteracting fermions in anisotropic harmonic confinement

Patrizia Vignolo, Anna Minguzzi

Published May 1, 2003• 21 citations
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Abstract

We develop a Green's function method to evaluate the exact equilibrium particle-density profiles of noninteracting Fermi gases in external harmonic confinement in any spatial dimension and for arbitrary trap anisotropy. While in a spherically symmetric configuration the shell effects are negligible in the case of a large number of particles, we find that for very anisotropic traps the quantum effects due to single-level occupancy and the deviations from the Thomas-Fermi approximation are also visible for mesoscopic clouds.

DOI
10.1103/physreva.67.053601

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Closed-form analytic expressions are derived for the density profile of a harmonically trapped noninteracting Fermi gas in ddd dimensions. Shell structure effects are included to leading order in 1/N1/N1/N, where NNN is the number of particles. These corrections to the local density approximation scale as $\ensuremath{\delta}n/n\ensuremath{\sim}{N}^{\ensuremath{-}\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=(1+1/d)/2$.
We present a Green's-function method for the evaluation of the particle density profile and of the higher moments of the one-body density matrix in mesoscopic systems containing a large number N of Fermi particles moving independently on a line under an external potential. The usefulness of the method is illustrated by applications to a Fermi gas confined in a harmonic potential well, for which we evaluate the momentum flux density, entering the equations of generalized hydrodynamics, and the kinetic energy density, which is relevant to the density-functional theory. As a further application of the method we explicitly display the quantum mean square fluctuations of these quantities. We also study some properties of the kinetic energy functional Ekin[n(x)]{E}_{\mathrm{kin}}[n(x)]Ekin​[n(x)] in the same system. Whereas a local approximation to the kinetic energy density is demonstrably wrong, an exact single-valued relationship between the density derivative of Ekin[n(x)]{E}_{\mathrm{kin}}[n(x)]Ekin​[n(x)] and the particle density n(x)n(x)n(x) is demonstrated and evaluated for various values of the number of particles in the system.
We evaluate the scattering functions of a gas of spin-polarized, noninteracting fermions confined in a quasi-one-dimensional harmonic trap at zero temperature. The main focus is on the inelastic scattering spectrum and on the angular distribution of scattered light from a mesoscopic atomic cloud as probes of its discrete quantum levels and of its shell structure in this restricted geometry. The dynamic structure factor is calculated and compared with the results of a local-density approximation exploiting the spectrum of a one-dimensional homogeneous Fermi gas. The elastic and inelastic contributions to the static structure factor are separately evaluated: the inelastic term becomes dominant as the momentum transfer increases, masking a small peak at twice the Fermi momentum in the elastic term but still reflecting the discrete quantum levels of the cloud. A fast and accurate numerical method is developed to evaluate the full pair distribution function for mesoscopic inhomogeneous clouds.
We consider a system of one-dimensional non-interacting fermions in external harmonic confinement. Using an efficient Green's function method we evaluate the exact profiles and the pair correlation function, showing a direct signature of the Fermi statistics and of the single quantum-level occupancy. We also study the dynamical properties of the gas, obtaining the spectrum both in the collisionless and in the collisional regime. Our results apply as well to describe a one-dimensional Bose gas with point-like hard-core interactions.