Single-particle excitations in a trapped gas of Fermi atoms in the BCS-BEC 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 Delta(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 (E-g) 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(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(omega) gives information about Delta(r=0) at the center of the trap (which is comparable to the Fermi energy epsilon(F) in the crossover region). We show that I(omega) is very dependent on the spatial profile of the pair potential Delta(r) that is used. We also emphasize that the narrow "unpaired atom" peak in the rf data gives information about E-g and the low-energy (<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 Torma and co-workers. The LDA, while useful, can lead to an incorrect physical picture of the low-density surface region of the Fermi superfluid.