content_image/372805.jpg content_image/372810.jpg
<img src='content_image/372805.jpg'>

where I is the identity matrix. The molecular excitation is gapless in the long-wavelength limit, k = 0 and ω = 0, following Eq. (4). At small k and ω , to the leading order of k , the molecular excitation frequency is linearly dispersed. The detail of molecular excitations will be studied in our future work. 

## C. Binding energy 

The binding energy E$_{b}$ of a Feshbach molecule in the molecular-BEC state is defined as the energy difference between two atomic and one molecular excitations at k = 0. Since the molecular excitation energy is gapless in the molecular-BEC state, the binding energy of Feshbach molecules is twice the energy gap of atomic excitations, E$_{b}$ = 2 E$_{0}$ . Due to many-body effect, the binding energy E$_{b}$ is different from its vacuum value E$_{b}$$_{0}$ = ¯ h $^{2}$/ ( ma 2 $_{s}$). At the superfluid transition temperature T$_{p}$ , the binding energy is given by E$_{b}$ ( T$_{p}$ ) = − 2 µ $^{′}$. In the limit of weakly-attractive interaction, n 1 / $^{3}$a$_{s}$ → 0 $^{−}$, we obtain 

<img src='content_image/372810.jpg'>

where ζ ( x ) is the Riemann-zeta function and T$_{a}$ is the ideal BEC temperature, T$_{a}$ = (2 π ¯ h $^{2}$/mk$_{B}$ )[ n/ζ (3 / 2)] 2 / $^{3}$; in the limit of weakly repulsion, n 1 / $^{3}$a$_{s}$ → 0 $^{+}$, the binding en- 

