by a single-channel model, 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 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, 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, Δ = − g V ∑ k u$_{k}$v$_{k}$ [1 + 2 f$_{k}$ ] , E$_{0}$ = 1 2 ∑ k [ E$_{k}$ − ϵ$_{k}$ + µ ′ + Δ 2 2 E$_{k}$ coth( βE$_{k}$ 2 ) ] − gn $^{2}$V, H$_{p}$ = E$_{0}$ + ∑ k E$_{k}$α † $_{k}$α$_{k}$ , (3) H$_{p}$ = H 0 p − ¯ h 2 2 m ψ $^{†}$∇ $^{2}$ψ + (2 gn − µ ) ψ $^{†}$ψ + Δ 2 ( ψ $^{†}$ψ † + ψψ ) , (2) H = − ¯ h 2 2 m ψ $^{†}$∇ $^{2}$ψ + g 2 ψ $^{†}$ψ $^{†}$ψψ − µψ $^{†}$ψ, (1) 3