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arXiv:cond-mat/0510296 · Modern Physics Letters B

ABSENCE OF BCS CONDENSATION OF ATOMS IN A REPULSIVE FERMI GAS OF ATOMS

Fuxiang Han, HAILING LI, Minghao Lei, E. WU

Published Jan 9, 2008• 0 citations
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Abstract

Bose–Einstein condensation of molecules and BCS condensation of Cooper pairs of atoms in an ultra-cold Fermi gas of atoms with a repulsive two-body interaction are studied by using the path integral representation of the grand partition function. From the self-consistent equations obtained in the present work for the order parameters, we have found that BCS condensation of atoms cannot occur in such a Fermi gas of atoms and that the condensate observed experimentally is composed of condensed molecules or possibly of preformed Cooper pairs. To substantiate our conclusions from the self-consistent equations for the order parameters, the effective atom–atom interaction mediated through molecules has also been computed with the Foldy–Wouthuysen transformation and has been found to be repulsive, which implies a net repulsive two-body interaction between atoms and hence the absence of BCS condensation of atoms.

DOI
10.1142/s0217984908014547

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Bose–Einstein condensation of molecules and BCS condensation of Cooper pairs of atoms in an ultra-cold Fermi gas of atoms with a repulsive two-body interaction are studied by using the path integral representation of the grand partition function. From the self-consistent equations obtained in the present work for the order parameters, we have found that BCS condensation of atoms cannot occur in such a Fermi gas of atoms and that the condensate observed experimentally is composed of condensed molecules or possibly of preformed Cooper pairs. To substantiate our conclusions from the self-consistent equations for the order parameters, the effective atom–atom interaction mediated through molecules has also been computed with the Foldy–Wouthuysen transformation and has been found to be repulsive, which implies a net repulsive two-body interaction between atoms and hence the absence of BCS condensation of atoms.
The self-consistent equations for the order parameters of Bose–Einstein condensation (BEC) of molecules and Bardeen–Cooper–Schrieffer (BCS) condensation of atoms in a Fermi gas of atoms with an attractive two-body interaction between atoms have been derived within the Hartree–Fock–Bogoliubov approximation from the path integral representation of the grand partition function. We have found that the order parameters for BEC and BCS are proportional to each other, which implies that BEC and BCS onsets simultaneously. We have also found that the common critical temperature of BEC and BCS increases as the average number of molecules increases and that the atom-molecule coupling enhances the common critical temperature.
This study presents a simple idea and a conceptual understandingof the crossover from the Bardeen Cooper (BCS) state of weakly-correlatedpairs of fermions to the Bose{Einstein condensation (BEC) of diatomicmolecules in the atomic Fermi gas. In the context of ultra cold Fermi gases,a BCS-BEC crossover means that by tuning the interaction strength, onegoes from a BCS state to a BEC state without encountering a phase transition.BEC state is a Bose-Einstein condensate of two-atom molecules (boundfermions), while the BCS state is made up of pair of atoms. The differencebetween the pairs and the molecules is that the molecules are localized inthe real space, whereas the BCS pairs are made of two particles with oppositemomenta in the momentum space.
We investigate within a self-consistent theory the molecular instabilities arising in the normal state of a homogeneous degenerate Fermi gas, covering the whole Bose-Einstein condensate (BEC) to BCS crossover. These are the standard instability for molecular formation, the BCS instability which corresponds to the formation of Cooper pairs, and the related Bose-Einstein instability. These instabilities manifest themselves in the properties of the particle-particle vertex, which we calculate in a ladder approximation. To find the critical temperatures corresponding to these various instabilities, we handle the properties of the interacting Fermi gas on the same footing as the instabilities by making use of the same vertex. This approximate treatment is shown to be quite satisfactory in a number of limiting situations where it agrees with known exact results. The results for the BCS critical temperature and for the BE condensation are found to be in fair agreement with earlier results. The threshold for formation of molecules at rest undergoes a sizable shift toward the BEC side, due to quantum effects arising from the presence of the degenerate Fermi gas. This should make its experimental observation fairly easy. This shift remains important at least up to temperatures comparable to the Fermi energy of the gas.
We investigate the Bose-Einstein condensation of fermionic pairs in a uniform two-component Fermi gas, obtaining an explicit formula for the condensate density as a function of the chemical potential and the energy gap. We analyze the condensate fraction in the crossover from the Bardeen-Cooper-Schrieffer (BCS) state of weakly interacting Cooper pairs to the Bose-Einstein condensate of molecular dimers. By using the local-density approximation we study confined Fermi vapors of alkali-metal atoms for which there is experimental evidence of condensation also on the BCS side of the Feshbach resonance. Our theoretical results are in agreement with these experimental data and give the behavior of the condensate on both sides of the Feshbach resonance at zero temperature.
We discuss the effect of a molecular Bose condensate on the energy of Fermi excitations in a trapped two-component atomic Fermi gas. The single-particle Green's functions can be approximated by the well-known BCS form, in both the BCS (Cooper pairs) and BEC (Feshbach resonance molecules) domains. The composite Bose order parameter Δ~{\tilde Δ}Δ~ describing bound states of two atoms and the Fermi chemical potential μμμ are calculated self-consistently. In the BEC regime characterized by μ<0μ<0μ<0, the Fermi quasiparticle energy gap is given by μ2+Δ~2\sqrt{μ^2+{\tilde Δ}^2}μ2+Δ~2​, instead of ∣Δ~∣|{\tilde Δ}|∣Δ~∣ in the BCS region, where μ>0μ>0μ>0. This shows up in the characteristic energy of atoms from dissociated molecules.
A self-consistent theory is derived to describe the BCS--Bose-Einstein-condensate crossover for a strongly interacting Fermi gas with a Feshbach resonance. In the theory the fluctuation of the dressed molecules, consisting of both preformed Cooper pairs and ``bare'' Feshbach molecules, has been included within a self-consistent TTT-matrix approximation, beyond the Nozi\`eres and Schmitt-Rink strategy considered by Ohashi and Griffin. The resulting self-consistent equations are solved numerically to investigate the normal-state properties of the crossover at various resonance widths. It is found that the superfluid transition temperature Tc{T}_{c}Tc​ increases monotonically at all widths as the effective interaction between atoms becomes more attractive. Furthermore, a residue factor Zm{Z}_{m}Zm​ of the molecule's Green function and a complex effective mass have been determined to characterize the fraction and lifetime of Feshbach molecules at Tc{T}_{c}Tc​. Our many-body calculations of Zm{Z}_{m}Zm​ agree qualitatively well with recent measurments of the gas of 6Li^{6}\mathrm{Li}6Li atoms near the broad resonance at 834G834\phantom{\rule{0.3em}{0ex}}\mathrm{G}834G. The crossover from narrow to broad resonances has also been studied.