Harmonically trapped fermion gases: exact and asymptotic results in arbitrary dimensions

Brack, Matthias; Murthy, M. V. N.

doi:10.1088/0305-4470/36/4/318

Cited by 37 publications

(90 citation statements)

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“…Consequently we are able to derive simple analytic formulae. Near the center of the trap, our results agree with those derived by Gleisberg et al [5] and Brack and Murthy [6]. Our expressions match the exact density profile throughout the cloud, even for surprisingly small numbers of particles: N ∼ 1.…”

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

Mueller

2004

Phys. Rev. Lett.

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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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confidence: 90%

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

Mueller

2004

Phys. Rev. Lett.

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“…Let us recall some basic quantum-mechanical definitions, using the same notation as in [11]. We start from the stationary Schrödinger equation for particles with mass m, bound by a local potential V (r) with a discrete energy spectrum {E n }:…”

Section: Basic Quantum-mechanical Definitionsmentioning

confidence: 99%

“…Recent experimental success confining fermion gases in magnetic traps [2] has led to a renewed interest in theoretical studies of confined degenerate fermion systems at zero [3,4,5,6,7,8,9,10,11,12] and finite temperatures [13,14]. Quite some effort has been devoted in these articles to establish local virial theorems for various types of confining potentials.…”

Section: Introductionmentioning

confidence: 99%

“…Quite some effort has been devoted in these articles to establish local virial theorems for various types of confining potentials. In [11,14], exact local virial theorems have been established for fermions bound in isotropic harmonic oscillator (IHO) potentials in arbitrary space dimension D. Some alternative virial theorems involving differentiation or integration of their particle density were also given in [11]. Our aim here is to investigate to what extent the results of [11,14] may be generalised to arbitrary local potentials V (r).…”

Section: Introductionmentioning

confidence: 99%

“…In [11,14], exact local virial theorems have been established for fermions bound in isotropic harmonic oscillator (IHO) potentials in arbitrary space dimension D. Some alternative virial theorems involving differentiation or integration of their particle density were also given in [11]. Our aim here is to investigate to what extent the results of [11,14] may be generalised to arbitrary local potentials V (r). While an obvious attempt is to simply replace the IHO potential V (r) = c r 2 by an arbitrarily chosen local potential V (r) in all those relations, we can only show that this leads to exact results for the D-dimensional linear potential V (r) = a · r with a constant vector a (which is not confining, but whose densities can nevertheless be calculated).…”

Section: Introductionmentioning

confidence: 99%

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Exact and asymptotic local virial theorems for finite fermionic systems

Brack

Koch

Murthy

et al. 2010

J. Phys. A: Math. Theor.

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Abstract.We investigate the particle and kinetic-energy densities for a system of N fermions confined in a potential V (r). In an earlier paper [J. Phys. A: Math. Gen. 36, 1111(2003], some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point r, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper we show that these local virial theorems (LVT) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are suggested by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although they formally are only valid asymptotically for large particle numbers N , we show that they practically are surprisingly accurate also for moderate values of N .

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Semiclassical Results for Ideal Fermion Systems. A Review

Combescure

Methods of Spectral Analysis in Mathematical Physics

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Harmonically trapped fermion gases: exact and asymptotic results in arbitrary dimensions

Cited by 37 publications

References 17 publications

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

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

Exact and asymptotic local virial theorems for finite fermionic systems

Semiclassical Results for Ideal Fermion Systems. A Review

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