Broad vs. narrow Fano-Feshbach resonances in the BCS-BEC crossover with trapped Fermi atoms

Abstract

Abstract. -With reference to the broad and narrow Fano-Feshbach resonances of 6 Li at about 822G and 543G, we show that for the broad resonance a molecular coupled-channel calculation can be mapped with excellent accuracy onto an effective single-channel problem with a contact interaction. This occurs for a wide enough range of the magnetic field, that the full BCS-BEC crossover can be realized with a typical trap. For the narrow resonance, the mapping onto a single-channel problem and the realization of the BCS-BEC crossover are restricted to too narrow a range of the magnetic field to obtain them in practice. A general criterion is also formulated for deciding whether the BCS-BEC crossover can be exhausted within the single-channel model for any specific Fano-Feshbach resonance. In this way, the BCS-BEC crossover for Fermi atoms with the broad resonance is placed on the same footing as the corresponding crossover for different physical systems. In this context, a large amount of work has been made by adopting a contact potential for the effective fermion-fermion attraction, regularized in terms of the scattering length a F [12][13][14]. The use of this potential considerably simplifies the many-body diagrammatic structure, both in the normal [13,14] and broken-symmetry [13,15] phases. By this approach, only fermionic degrees of freedom are retained in the many-body Hamiltonian, in the same spirit of the original BCS theory [16]. Since just the quantity a F is varied in a controlled way by sweeping the magnetic field across a FF resonance, it would appear that trapped Fermi gases constitute an ideal testing ground for many-body theories based on the above two-body interaction. Previous theoretical work using the same interaction could thus be adapted to trapped Fermi gases with limited effort.The use of the same effective two-body interaction for such different systems like hightemperature superconductors and trapped Fermi gases may also lead to the emergence of universal features. This should be especially desirable, as the insights gleaned from the c EDP Sciences

Thermodynamics of Ultracold Fermi Gases

“…Its value r e ∼ 4.7 nm is very small for 6 Li gases [206,207]. The p-wave scattering length a p = 1.8 nm [208] describes low-energy p-wave interactions.…”

Thermodynamics of the BCS-BEC crossover

“…As has been shown for instance by Diener and Ho [11], however, the situation can be simplified in the case of broad Feshbach resonances, where the effective range r ⋆ of the resonant interaction is much smaller than both the background scattering length a bg and the Fermi wavelength λ F . In this limit, which is in fact appropriate for the existing experimental studies of the BCS-BEC crossover problem in 6 Li [5] and in 40 K [1], the problem can be reduced to a single channel Hamiltonian with an instantaneous interaction [10,11,12,13]. The associated effective two-body interaction is thus described by a pseudopotential V (r) ∼ δ(r) (appropriately renormalized, see below) with a strength proportional to the scattering length…”

“…A particularly challenging problem arises right at the Feshbach resonance, where the two-particle scattering length is infinite [8,9]. Precisely at this point and for broad Feshbach resonances, where the range r ⋆ of the effective interaction is much smaller than the mean interparticle spacing [10,11,12,13], the full many-body problem has the Fermi energy ε F as the only energy scale. As pointed out by Ho [14], the thermodynamics of the unitary Fermi gas is then a function only of the dimensionless temperature θ = T /T F .…”

Multiscale quantum-defect theory for two interacting atoms in a symmetric harmonic trap

“…(19). If the scattering length parameter used in our formulation is achieved by tuning around a Feshbach resonance [36,37], the same results would apply, provided it is a broad Feshbach resonance with a width much greater than the energy scale s E associated with the van der Waals potential [38,39,40].…”

“…It uses the representation of Eq. (39) to facilitate comparison with the shape-independent approximation (see Ref. [2] and Sec.…”