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arXiv:0906.1239 · Europhysics Letters (EPL)

On the kinetic energy of unitary Fermi gas in a harmonic trap

A. L. Zubarev, M. Zoubarev

Published Aug 1, 2009• 5 citations
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Abstract

We have considered the orbital-free approximation of the kinetic energy functional to investigate the zero-temperature properties of dilute harmonically trapped two-component Fermi gas at unitarity. It is shown that our approach provides a realible and inexpensive method to study superfluid strongly interacting dilute Fermi gases.
DOI
10.1209/0295-5075/87/33001

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We have considered the orbital-free approximation of the kinetic energy functional to investigate the zero temperature properties of dilute harmonically trapped two component Fermi gas at unitarity. It is shown that our approach provides a realible and inexpensive method to study superfluid strongly interacting dilute Fermi gases. PACS numbers: 31.15.Ew, 71.15.Mb, 03.75.SsThe modern density functional theory (DFT) is based on the Kohn-Sham (KS) approach [1], where the noninteracting kinetic energy, T, is calculated in terms of the KS orbitals, although Ref.[2] proved the basic existence of functional T(ρ), where ρ is the density, ρ(⃗r)d
Strongly interacting, dilute Fermi gases exhibit a scale-invariant, universal thermodynamic behavior. This is notoriously difficult to understand theoretically because of the absence of a small interaction parameter. Here, we present a systematic comparison of theoretical predictions from different quantum many-body theories with recent experimental data of Nascimbne et al (2010 Nature 463 1057). Our comparisons have no adjustable parameters, either theoretically or experimentally. All the model approximations seem to fluctuate rather than converge on the experimental data. It turns out that a simple Gaussian pair fluctuation theory gives the best quantitative agreement, except at the critical superfluid transition region. In the normal state, we also calculate the equation of state by using a quantum cluster expansion theory and explore in detail its applicability to low temperatures. Using the accurate experimental result for the thermodynamic function S ( T ), we determine the temperature T of a trapped Fermi gas at unitarity as a function of a non-interacting temperature T i , which can be obtained by an adiabatic sweep to the free gas limit. By analyzing the recent experimental data, we find a characteristic temperature ( T / T F ) 0 =0.19±0.02 or ( T i / T F ) 0 =0.16±0.02 in a harmonic trap, below which there are deviations from normal Fermi-liquid-like behavior that may be attributed to pairing effects. Here, T F is the Fermi temperature for a trapped ideal, non-interacting Fermi gas. Our thorough comparison may shed light on the further theoretical development of strongly interacting fermions.
The zero-temperature properties of a dilute two-component Fermi gas in the BCS to Bose-Einstein Condensation crossover are investigated. On the basis of a generalization of the Hylleraas-Undheim method, we construct rigorous upper bounds to the collective frequencies for the radial and the axial breathing mode of the Fermi gas under harmonic confinement in the framework of the hydrodynamic theory. The bounds are compared to experimental data for trapped vapors of 6Li^{6}\mathrm{Li}6Li atoms.
We examine a dilute two-component atomic Fermi gas trapped in a harmonic potential in the superfluid phase. For experimentally realistic parameters, the trapping potential is shown to have crucial influence on various properties of the gas. Using an effective Hamiltonian, analytical results for the critical temperature, the temperature dependence of the superfluid gap, and the energy of the lowest collective modes are derived. These results are shown to agree well with numerical calculations. We also discuss in more detail a previously proposed method to experimentally observe the superfluid transition by looking at the collective-mode spectrum. Our results are aimed at the present experimental effort to observe a superfluid phase transition in a trapped atomic Fermi gas.
We theoretically investigate the spin-dipole oscillation of a strongly interacting Fermi gas confined by a harmonic trapping potential. By using a diagrammatic strong-coupling theory combined with a local density approximation and a sum rule approach, we study the temperature dependence of the spin-dipole frequency near unitarity. The connection of the spin-dipole frequency with the spin susceptibility and the pairing correlations is exploited. While the spin-dipole frequency exactly coincides with the trap frequency in a noninteracting Fermi gas, it is shown to be strongly enhanced in the superfluid state, because of the suppression of the spin degree of freedom due to the spin-singlet Cooper-pair formation. In strongly interacting Fermi gases, such enhancement occurs even above the superfluid phase transition temperature, due to the strong pairing correlations.
The recently discovered universal thermodynamic behavior of dilute, strongly interacting Fermi gases also implies a universal structure in the many-body pair-correlation function at short distances, as quantified by the contact . Here, we theoretically calculate the temperature dependence of this universal contact for a Fermi gas in free space and in a harmonic trap. At high temperatures above the Fermi degeneracy temperature, T ≳ T F , we obtain a reliable non-perturbative quantum virial expansion up to third order. At low temperatures, we compare different approximate strong-coupling theories. These make different predictions, which need to be tested either by future experiments or by advanced quantum Monte Carlo simulations. We conjecture that in the universal unitarity limit, the contact or correlation decreases monotonically with increasing temperature, unless the temperature is significantly lower than the critical temperature, T ≪ T c ∼0.2 T F . We also discuss briefly how to measure the universal contact in either homogeneous or harmonically trapped Fermi gases.