III. MECHANICAL COLLAPSE OF COHERENT ATOM-MOLECULE MIX- TURE At the lower boundary of the molecular-BEC phase, the atomic excitation energy van- ishes, E$_{0}$ = 0, suggesting the appearance of the coherent atom-molecule mixture. In the mixture, since both molecules and atoms exhibit ODLRO, two order parameters can be intro- duced [12], ψ$_{0}$ ≡ 〈 ψ 〉 describing atomic condensate and Δ ≡ g 〈 δψδψ 〉 describing molecular condensate, where δψ ≡ ψ − ψ$_{0}$ . These two order parameters are in principle independent, whereas in the atomic-BEC state Δ is a function of ψ$_{0}$ . In the mean-field approximation, the Hamiltonian density for this mixture state is given by H$_{m}$ = H 0 m − ¯ h 2 2 m δψ $^{†}$∇ $^{2}$δψ + (2 gn − µ ) δψ $^{†}$δψ + 1 2 [ (Δ + gψ 2 $_{0}$) δψ $^{†}$δψ † + h.c. ] , (16) where H 0 m = gn 2 $_{0}$/ 2 − µn$_{0}$ − gδn 2 − | Δ | $^{2}$/ 2 g , n$_{0}$ ≡ | ψ$_{0}$ | 2 is the atomic-condensate density, δn ≡ 〈 δψ $^{†}$δψ 〉 , and n = n$_{0}$ + δn . The order parameter ψ$_{0}$ should minimize the mean-field thermodynamic potential, which leads to the saddle point condition To satisfy the saddle point condition, the product Δ ψ 2 0 must be real. For simplicity, we choose Δ > 0. µ = g ( n$_{0}$ + 2 δn ) + Δ ψ ∗ 0 ψ$_{0}$ . (17) As in the molecular BEC case, the mean-field Hamiltonian of the mixture can be diago- nalized by Bogoliubov transformation, where the field operator of atomic quasi-particles is given by α$_{k}$ = u$_{k}$ψ$_{k}$ + v$_{k}$ψ † − $_{k}$, u 2 k = v 2 $_{k}$+1 = [1+( ϵ$_{k}$ − µ $^{′}$) /E$_{k}$ ] / 2, and the quasi-particle energy is given by E$_{k}$ = √ ( ϵ$_{k}$ − µ $^{′}$) 2 − ( gψ 2 0 + Δ) $^{2}$. The order parameter Δ can be determined by the self-consistency equation where similar to Eq. (4) the last term in the integrand is the counter term. The equation for the total density is given by i.e. Δ = − g V ∑ k u$_{k}$v$_{k}$ [1 + 2 f$_{k}$ ] , − m 4 π ¯ h $^{2}$a$_{s}$ + ψ 2 0 Δ + gψ 2 0 = ∫ d $^{3}$k (2 π ) 3 [ 1 2 E$_{k}$ coth ( βE$_{k}$ 2 ) − 1 2 ϵ$_{k}$ ] , (18) n = n$_{0}$ + ∫ d $^{3}$k (2 π ) 3 [ ϵ$_{k}$ − µ ′ 2 E$_{k}$ coth ( βE$_{k}$ 2 ) − 1 2 ] . (19) 10