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arXiv:1002.3270

Cold atoms at unitarity and semiclassical ground state of a Fermi gas for Haldane-Wu exclusion statistics

Jérôme Roccia

Published Feb 17, 2010• 0 citations
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Abstract

We investigate finite particle systems of cold atoms bound in a local potential V(r). We derive the ground state energy and the particle density using a recently developed semiclassical theory (2008 Phys. Rev. Lett. 100 200408), and assuming the particles are described by the Haldane-Wu fractional exclusion statistics (FES) at unitarity. This approach is applied to atoms trapped into a three dimensional harmonic oscillator. We show that the parameter-free FES semiclassical theory yields results that are consistent with numerical simulations by Chang and Bertsch [2007 Phys. Rev A 76 021603(R)] and Bulgac (2007 Phys. Rev. A 76 040502).

DOI
10.48550/arxiv.1002.3270

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We consider a few-particle system of trapped neutral fermionic atoms at ultra-low temperatures, with the attractive interaction tuned to Feshbach resonance. We calculate the energies and the spatial densities of the few-body systems using a generalization of the extended Thomas–Fermi (ETF) method, and assuming the particles obey the Haldane–Wu fractional exclusion statistics (FES) at unitarity. This method is different from the scaled ETF version given by Chang and Bertsch (2007 Phys. Rev. A 76 021603). Our semiclassical FES results are consistent with the Monte Carlo calculations of the above authors, but can hardly be distinguished from their overall scaling of the ETF result at unitarity.
We consider N fermions in a two-dimensional harmonic oscillator potential interacting with a very short-range repulsive pair-wise potential. The ground-state energy of this system is obtained by performing a Thomas-Fermi as well as a self-consistent Hartree-Fock calculation. The two results are shown to agree even for a small number of particles. We next use the finite-temperature Thomas-Fermi method to demonstrate that in the local density approximation, these interacting fermions are equivalent to a system of noninteracting particles obeying the Haldane-Wu fractional exclusion statistics. It is also shown that mapping onto a system of N noninteracting quasiparticles enables us to predict the energies of the ground and excited states of the N-body system. PACS Nos.: 05.30-d, 73.20Dx
Consider two identical atoms in a spherical harmonic oscillator interacting with a zero-range interaction which is tuned to produce an s-wave zero-energy bound state. The quantum spectrum of the system is known to be exactly solvable. We note that the same partial wave quantum spectrum is obtained by the one-dimensional scale-invariant inverse square potential. Long known as the Calogero–Sutherland–Moser (CSM) model, it leads to the fractional exclusion statistics (FES) of Haldane and Wu. The statistical parameter is deduced from the analytically calculated second virial coefficient. When FES is applied to a Fermi gas at unitarity, it gives good agreement with experimental data without the use of any free parameter.
We consider a gas of neutral fermionic atoms at ultra-low temperatures, with the attractive interaction tuned to Feshbach resonance. We calculate the variation of the chemical potential and the energy per particle as a function of temperature by assuming the system to be an ideal gas obeying the Haldane–Wu fractional exclusion statistics. Our results for the untrapped gas compare favourably with the recently published Monte Carlo calculations of two groups. For a harmonically trapped gas, the results agree with the experiment, and also with other published works.
We consider N fermions in a two-dimensional harmonic oscillator potential interacting with a very short-range repulsive pair-wise potential. The ground-state energy of this system is obtained by performing a Thomas-Fermi as well as a self-consistent Hartree-Fock calculation. The two results are shown to agree even for a small number of particles. We next use the finite temperature Thomas-Fermi method to demonstrate that in the local density approximation, these interacting fermions are equivalent to a system of noninteracting particles obeying the Haldane-Wu fractional exclusion statistics. It is also shown that mapping onto to a system of N noninteracting quasi-particles enables us to predict the energies of the excited states of the N-body system.
We utilize a fractional exclusion statistics of Haldane and Wu hypothesis to study the thermodynamics of a unitary Fermi gas trapped in a harmonic oscillator potential at ultra-low finite temperature. The entropy per particle as a function of the energy per particle and energy per particle versus rescaled temperature are numerically compared with the experimental data. The study shows that, except the chemical potential behavior, there exists a reasonable consistency between the experimental measurement and theoretical attempt for the entropy and energy per particle. In the fractional exclusion statistics formalism, the behavior of the isochore heat capacity for a trapped unitary Fermi gas is also analyzed.