 Normal-ordered boson pair count

For a bosonic mode satisfying the canonical commutation relation $[a,a^{†}]=1$, the number operator $n=a^{†}a$ obeys $n^2=a^{†}a{a}^{†}a=(a^{†}{)}^2{a}^2+n$. Thus a two-body on-site interaction is proportional to $n(n-1)$, with no self-interaction. In a fixed-number two-mode condensate with probability $q$ of occupying this mode, $⟨n(n-1)⟩=N(N-1)q^2$ by the binomial distribution.