content_image/373847.jpg content_image/373845.jpg content_image/373844.jpg content_image/373843.jpg content_image/373842.jpg content_image/373841.jpg
where f$_{k}$ is the Bose distribution function of quasi-particles, f$_{k}$ = 1 / [exp( βE$_{k}$ ) − 1]. This self-consistency equation can be written explicitly as 

<img src='content_image/373847.jpg'>

where the term − 1 / 2 ϵ$_{k}$ in the integrand on the right-hand side is a counter term due to vacuum renormalization. The chemical potential µ and the order parameter Δ can be solved from Eq. (4) and the following equation for the density n , 

## B. Excitations in the long-wavelength limit 

There is a gap in the atomic excitation energy given by 

<img src='content_image/373845.jpg'>

<img src='content_image/373844.jpg'>

If the molecular-BEC state is stable, one necessary condition is that the gap E$_{0}$ is real, or − µ ′ ≥ Δ. When − µ ′ = Δ, the atomic excitation becomes gapless, E$_{0}$ = 0, which marks the lower boundary of the molecular-BEC phase. The upper phase boundary is located at the superfluid transition temperature T$_{p}$ where Δ = 0 and E$_{0}$ = − µ $^{′}$. 

In addition to atomic excitations, there are collective excitations of Feshbach molecules. Although the molecular excitation energy cannot be obtained in the mean-field approxima- tion, it can be extracted from poles of two-particle correlation functions given by 

<img src='content_image/373843.jpg'>

where b$_{1}$ = ψ $^{2}$, b$_{2}$ = b † $_{1}$, b$_{3}$ = 2 ψ $^{†}$ψ , 0 ≤ τ ≤ β ¯ h , and T is the time-ordering operator. As in the zero-temperature case [12], the correlation function can be calculated in the random- phase approximation (RPA), 

<img src='content_image/373842.jpg'>

where χ $^{(0)}$( k , ω ) is the correlation function calculated in the mean-field approximation. In RPA, the dispersion of Feshbach molecules satisfies the following equation, 

<img src='content_image/373841.jpg'>

