content_image/367819.jpg content_image/367818.jpg content_image/367817.jpg content_image/367816.jpg content_image/367815.jpg
by a single-channel model, 

<img src='content_image/367819.jpg'>

where ψ is the field operator of Bose atoms in the open channel, the coupling constant is given by g = 4 π ¯ h $^{2}$a$_{s}$ /m , and µ is the chemical potential of atoms. 

In the molecular-BEC state, atoms are paired into molecules and the off-diagonal long range order (ODLRO) appears, Δ ≡ g 〈 ψψ 〉 . Without losing any generality, Δ ≥ 0 is assumed in the following. In the mean field, in addition to the pairing energy, both Hartree and Fock energies should be taken into account, and the mean-field Hamiltonian density is given by 

<img src='content_image/367818.jpg'>

where the constant H 0 p is given by H 0 p = − ( gn $^{2}$+ Δ$^{2}$/ 2 g ). In this mean-field approximation, the pairing gap Δ and Hartree-Fock energy are determined self-consistently, but the fluctua- tion and three-body effects are ignored. The mean-field theory provides a description about single-particle excitations, but it is incapable to describe collective excitations or few-body properties such as Effimov effect. Nonetheless, the mean-field approximation can serve as a starting point of the theoretical description for the molecular-BEC state. 

The mean-field Hamiltonian can be diagonalized by Bogoliubov transformation, 

<img src='content_image/367817.jpg'>

where α$_{k}$ = u$_{k}$ψ$_{k}$ + v$_{k}$ψ † − k is the annihilation operator of atomic quasi-particles, E$_{k}$ = √ ( ϵ$_{k}$ − µ $^{′}$) 2 − Δ 2 is the quasi-particle energy, and the transformation coefficients are given by u 2 k = v 2 k + 1 = [1 + ( ϵ$_{k}$ − µ $^{′}$) /E$_{k}$ ] / 2 with ϵ$_{k}$ = ¯ h $^{2}$k$^{2}$/ 2 m and µ ′ = µ − 2 gn . The energy constant E$_{0}$ is given by 

where 1 /β = k$_{B}$T and V is the volume. 

The order parameter Δ can be determined self-consistently, 

<img src='content_image/367816.jpg'>

<img src='content_image/367815.jpg'>

