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=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), 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.
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 at a magnetic field of
B0=834G and find good agreement with experiments in that case. The BEC-BCS crossover for more narrow Feshbach resonances is also discussed.
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
s-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
s-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.
We investigate the non-universal behavior of the BCS-BEC crossover model at the normal to superfluid transition. By using a finite temperature quantum field theoretical approach due to Nozieres and Schmitt-Rink and by making the effective range expansion of the effective two-body interaction we numerically calculate the crossover transition temperature as a function of the scattering length,
aF, and the effective range parameter,
re. In an ultracold two-component atomic Fermi gas BCS-BEC crossover physics is expected to appear near magnetic-field-induced Feshbach resonances. By matching two-body scattering properties near a Feshbach resonance to a simple renormalized model potential, a broad resonance is characterized by a small effective range and the gas displays a universal BCS-BEC crossover behavior. On the other hand, for a narrow resonance thermodynamic quantities depend the effective range which may be large and negative at resonance. For increasing values of
−re we find the transition temperature to be suppressed in the crossover region. We furthermore argue for existence of a lower bound of
Tc at fixed coupling for increasing
−re in the crossover region.
We investigate the single-particle properties at
T=0 of a trapped superfluid gas of Fermi atoms with a Feshbach resonance. A tunable pairing interaction associated with the Feshbach resonance leads to the BCS--Bose-Einstein condensate (BEC) crossover, where the character of superfluidity continuously changes from the BCS-type to a BEC of composite bosons. In this paper, we extend our previous work for a uniform superfluid Fermi gas [Y. Ohashi and A. Griffin, Phys. Rev. A 67, 063612 (2003)] to include the effect of a harmonic trap. We do not use the local density approximation (LDA), but directly solve the Bogoliubov--de Gennes (BdG) coupled equations. While our explicit numerical solutions are for a weak (narrow) Feshbach resonance, we argue that the single-particle BdG excitation spectrum will exhibit the same features for a strong (broad) Feshbach resonance. Using these equations, we find self-consistent values for the spatially dependent local density
n(r) as well as the composite BCS order parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(\mathbf{r})$, the latter describing both the Cooper-pair and molecular condensate contributions. Using these results, we calculate the single-particle density of states in the crossover region, and from this determine the true single-particle energy gap
(Eg) of the trapped Fermi superfluid. This is associated with the in-gap (or Andreev) states in the low-density region at the edge of the trap. We calculate the laser-induced tunneling current $I(\ensuremath{\omega})$ into another hyperfine state, as measured in recent rf spectroscopy experiments. This rf spectrum gives a direct probe of the quasiparticle spectrum. We show how the high-energy part of $I(\ensuremath{\omega})$ gives information about $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r=0)$ at the center of the trap (which is comparable to the Fermi energy ${\ensuremath{\epsilon}}_{F}$ in the crossover region). We show that $I(\ensuremath{\omega})$ is very dependent on the spatial profile of the pair potential $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}(r)$ that is used. We also emphasize that the narrow ``unpaired atom" peak in the rf data gives information about
Eg and the low-energy $(⪡{\ensuremath{\epsilon}}_{F})$ in-gap states of a Fermi superfluid. While our calculations are limited to
T=0, we use them to discuss the recent data of Chin et al. and the LDA calculations of T\"orm\"a and co-workers. The LDA, while useful, can lead to an incorrect physical picture of the low-density surface region of the Fermi superfluid.
We systematically study the BCS-BEC crossover and the quantum phase transition in ultracold
6Li and
40K atoms across a wide Feshbach resonance. The background scattering lengths for
6Li and
40K have opposite signs, which lead to very different behaviors for these two types of atoms. For
40K, both the two-body and the many-body calculations show that the system always has two branches of solutions: one corresponds to a deeply bound molecule state; and the other, the one accessed by the current experiments, corresponds to a weakly bound state with population always dominantly in the open channel. For
6Li, there is only a unique solution with the standard crossover from the weakly bound Cooper pairs to the deeply bound molecules as one sweeps the magnetic field through the crossover region. Because of this difference, for the experimentally accessible state of
40K, there is a quantum phase transition at zero temperature from the superfluid to the normal Fermi gas at the positive detuning of the magnetic field where the
s-wave scattering length passes its zero point. For
6Li, however, the system changes continuously across the zero point of the scattering length. For both types of atoms, we also give a detailed comparison between the results from the two-channel and the single-channel model over the whole region of the magnetic field detuning.