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arXiv:cond-mat/0609253 · Physical Review A

Exact coherent states of a noninteracting Fermi gas in a harmonic trap

Dae-Yup Song

Published Nov 16, 2006• 6 citations
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Abstract

Exact and closed-form expressions of the particle density and the momentum distribution are derived for a coherent state of a noninteracting Fermi gas, while such a state can be obtained from the ground state in a $d$-dimensional isotropic harmonic trap by modulating the trap frequency and shifting the trap center. The profile of the momentum distribution turns out to be identical in shape with that of the particle density, however, as a manifestation of the Heisenberg uncertainty principle, the dispersion of the distribution increases (decreases) when that of the particle density is decreased (increased). The expressions are also applicable for a sudden and total opening of the trap, and it is shown that, after the opening, the gas has a stationary momentum distribution whose dispersion could be arbitrarily large or small.

DOI
10.1103/physreva.74.051602

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Exact and closed-form expressions of the particle density and the momentum distribution are derived for a coherent state of a noninteracting Fermi gas, while such a state can be obtained from the ground state in a ddd-dimensional isotropic harmonic trap by modulating the trap frequency and shifting the trap center. The profile of the momentum distribution turns out to be identical in shape with that of the particle density, however, as a manifestation of the Heisenberg uncertainty principle, the dispersion of the distribution increases (decreases) when that of the particle density is decreased (increased). The expressions are also applicable for a sudden and total opening of the trap, and it is shown that, after the opening, the gas has a stationary momentum distribution whose dispersion could be arbitrarily large or small.
One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution n0(z,N).{n}_{0}(z,N).n0​(z,N). For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near 2kF,{2k}_{F},2kF​, with kF{k}_{F}kF​ being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.
We derive simple analytical expressions for the particle density $\ensuremath{\rho}(r)$ and the kinetic energy density $\ensuremath{\tau}(r)$ for a system of noninteracting fermions in a ddd-dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation $\ensuremath{\tau}[\ensuremath{\rho}]$ using the exact $\ensuremath{\rho}(r)$ and show that it locally reproduces the exact kinetic energy density $\ensuremath{\tau}(r)$, including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of ${\ensuremath{\tau}}_{\mathrm{TF}}[\ensuremath{\rho}(r)]$ yields the exact total kinetic energy.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
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 study normal state properties of an interacting Fermi gas in an isotropic harmonic trap of arbitrary dimensions. We exactly calculate the first-order perturbation terms in the ground-state energy and chemical potential, and obtain simple analytic expressions of the total energy and chemical potential. At zero temperature, we find that the Thomas–Fermi approximation agrees well with the exact results for any dimension even though the system is small. In the high-temperature (classical) region, we find the interaction energy decreases in proportion to T − d /2 , where T is the temperature and d is the dimension of the system.