arXiv:physics/9807027 · Physical Review A

Interacting Fermi gas in a harmonic trap

Georg M. Bruun, Keith R. Burnett

Published Sep 1, 1998• 76 citations

Abstract

In view of ongoing experiments to trap ultracold spin-polarized ${}^{6}\mathrm{Li},$ we study various properties of an interacting Fermi gas in a harmonic trap, taking the discrete nature of the unperturbed harmonic trap levels into account exactly. As ${}^{6}\mathrm{Li}$ has a rather large and negative scattering length, we focus on the effects of the attractive atom-atom interaction on several thermodynamic properties, and on the momentum and density distributions. The dependence of the chemical potential, the specific heat, and the density and momentum distributions on the number of particles in the trap is obtained. We also calculate the energy of the gas. Comparison is made with results of a semiclassical calculation and with the properties of a noninteracting gas. We find that the effect of the interactions is rather large for realistic trap frequencies. Hence it is important to include these interactions in any quantitative predictions relevant for experiments.

DOI: 10.1103/physreva.58.2427

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arXiv:physics/9807027 · Physical Review A

Interacting Fermi gas in a harmonic trap

Georg M. Bruun, Keith R. Burnett

Published Sep 1, 1998• 76 citations

arXiv HTML PDF

Abstract

In view of ongoing experiments to trap ultracold spin-polarized

{}^{6}\mathrm{Li},

we study various properties of an interacting Fermi gas in a harmonic trap, taking the discrete nature of the unperturbed harmonic trap levels into account exactly. As

{}^{6}\mathrm{Li}

has a rather large and negative scattering length, we focus on the effects of the attractive atom-atom interaction on several thermodynamic properties, and on the momentum and density distributions. The dependence of the chemical potential, the specific heat, and the density and momentum distributions on the number of particles in the trap is obtained. We also calculate the energy of the gas. Comparison is made with results of a semiclassical calculation and with the properties of a noninteracting gas. We find that the effect of the interactions is rather large for realistic trap frequencies. Hence it is important to include these interactions in any quantitative predictions relevant for experiments.

DOI: 10.1103/physreva.58.2427

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arXiv:cond-mat/9710086Physical Review AFeb 1, 199893 citations

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arXiv:0706.3360Reviews of Modern PhysicsOct 2, 20081,997 citations

In view of ongoing experiments to trap ultracold spin-polarized

{}^{6}\mathrm{Li},

we study various properties of an interacting Fermi gas in a harmonic trap, taking the discrete nature of the unperturbed harmonic trap levels into account exactly. As

{}^{6}\mathrm{Li}

We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number

N

of atoms, harmonic oscillator length, deformation of the trap,

s

-wave scattering length) fix its thermodynamic behavior in the large

N

limit through two dimensionless scaling parameters. These are the reduced temperature

t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}

and the ratio $\ensuremath{\eta}$ between the

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature

{T}_{c}^{0}

of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.

(P<{P}_{c})

or a fully polarized normal gas

(P>{P}_{c})

, depending on the smallness of the spin polarization

P

, relative to a critical value

{P}_{c}

. In this work, we show how these two phase-separation phases evolve with increasing temperature, and thereby construct a finite-temperature phase diagram. For typical interactions, we find that the exotic FFLO phase survives below one-tenth of Fermi degeneracy temperature, which seems to be accessible in the current experiment. The density profile, equation of state, and specific heat of the polarized system have been calculated and discussed in detail. Our results are useful for the ongoing experiment at Rice University on the search for FFLO states in quasi-one-dimensional polarized Fermi gases.

{n}_{0}(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

{2k}_{F},

with

{k}_{F}

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.

s

-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.