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arXiv:cond-mat/0607045 · Physical Review A

Bogoliubov theory of Feshbach molecules in the BEC-BCS crossover

M. W. J. Romans, Henk T. C. Stoof

Published Nov 17, 2006• 14 citations
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Abstract

We present the Bogoliubov theory for the Bose-Einstein condensation of Feshbach molecules in a balanced Fermi mixture. Because the Bogoliubov theory includes (Gaussian) fluctuations, we can in this manner accurately incorporate both the two-body and many-body aspects of the BEC-BCS crossover that occurs near a Feshbach resonance. We apply the theory in particular to the very broad Feshbach resonance in atomic $^{6}\mathrm{Li}$ at a magnetic field of ${B}_{0}=834\phantom{\rule{0.3em}{0ex}}\mathrm{G}$ and find good agreement with experiments in that case. The BEC-BCS crossover for more narrow Feshbach resonances is also discussed.

DOI
10.1103/physreva.74.053618

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We present the Bogoliubov theory for the Bose-Einstein condensation of Feshbach molecules in a balanced Fermi mixture. Because the Bogoliubov theory includes (Gaussian) fluctuations, we can in this manner accurately incorporate both the two-body and many-body aspects of the BEC-BCS crossover that occurs near a Feshbach resonance. We apply the theory in particular to the very broad Feshbach resonance in atomic 6Li^{6}\mathrm{Li}6Li at a magnetic field of B0=834G{B}_{0}=834\phantom{\rule{0.3em}{0ex}}\mathrm{G}B0​=834G and find good agreement with experiments in that case. The BEC-BCS crossover for more narrow Feshbach resonances is also discussed.
We present the random phase approximation (RPA) theory of the Bose-Einstein-condensation-Bardeen-Cooper-Schrieffer crossover in an atomic Fermi gas near a Feshbach resonance that includes the relevant two-body atomic physics exactly. This allows us to determine the probability for the dressed molecules in the Bose-Einstein condensate to be in the closed channel of the Feshbach resonance and to compare with the recent experiments of Partridge et al. [95, 020404 (2005)10.1103/PhysRevLett.95.020404] with , who have measured the same quantity.
We apply the formulation developed in a recent paper [Y. Ohashi and A. Griffin, Phys. Rev. A 72, 013601 (2005)] for single-particle excitations in the BCS-BEC crossover to the case of a broad Feshbach resonance. At T=0T=0T=0, we solve the Bogoliubov--de Gennes coupled equations taking into account a Bose condensate of bound states (molecules). In the case of a broad resonance, the density profile n(r)n(r)n(r), as well as the profile of the superfluid order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r)$, are spatially spread out to the Thomas-Fermi radius, even in the crossover region. This order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r)$ suppresses the effects of low-energy Andreev bound states on the rf tunneling current. As a result, the peak energy in the rf spectrum is found to occur at an energy equal to the superfluid order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r=0)$ at the center of the trap, in contrast to the case of a narrow resonance, and in agreement with recent measurements. The local density approximation is found to give a good approximation for the rf-tunneling spectrum.
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.
We extend our previous investigations of fermionic condensation in broad Feshbach resonances by using the two-channel model developed for narrow Feshbach resonances. We investigate two crossovers: the Bardeen-Cooper-Schrieffer--Bose-Einstein condensation (BCS-BEC) crossover by changing the sss-wave scattering length and the crossover from a narrow to a broad resonance by changing the atom-molecule coupling. At zero temperature we analyze, as a function of both atom-molecule coupling and sss-wave scattering length, the chemical potential, the energy gap, and the condensate fraction of atoms. In particular, we predict the contribution of Cooper pairs and preformed molecules to the total condensate density along the two crossovers.
To study the behavior of atomic Fermi gases in the vicinity of Feshbach resonance, M. Holland et al. [12] and S. J. J. M. F. Kokkelmans et al. [22] proposed resonance mean field theory in which the molecules formed by Feshbach resonance are considered in the many-body Hamiltonian. Motivated by microscopic scattering theory and resonance mean field theory, we develop a thermodynamic method to study this system. In this method we treat atomic Fermi gases near resoance as a atom-molecule mixture. We can use BCS Hamiltonian and Bose- Einstein statistics to describe the behavior of atoms and molecules respectively. Chemical equilibrium can be achieved between atoms and molecules due to hypefine interaction. We use this method to study the system behavior under different temperature and detuning energy and find that at low temperature a smooth BCS-BEC crossever can occure with decreasing detuning energy. We further calculate a measurable quantity, molecular condensation fraction, and find its behavior matches qualitatively with the experimental result [16].
We consider a single vortex in a superfluid Fermi gas in the BCS-BEC crossover regime near a Feshbach resonance. The effect of the molecular Bose-Einstein condensate upon the vortex structure is discussed within the mean field approximation at zero temperature. Using the self-consistent Bogoliubov-de Gennes equation of the fermion-boson coupled model, we calculate density distributions of atoms and molecules. As the number of the molecules increases, both atomic and molecular density changes from BCS-like distribution to BEC-like. We also study the change of the vortex core size in the crossover regime.