One-body coherence of a Hartree condensate ($G^{(1)}(x,y)=N{\varphi }^{\ast }(x)\varphi (y)$)

For the normalized Hartree product state $|N:\varphi ⟩$, applying the bosonic annihilation operator gives $\psi (y)|N:\varphi ⟩=\sqrt{N}\varphi (y)|N-1:\varphi ⟩$. Taking the inner product proves the displayed factorization of the one-body density matrix. Its diagonal is the total number density in this approximation. For a real positive orbital, $G^{(1)}(0,r)=\sqrt{n(0)n(r)}$. Restoring the kinetic energy in a Gross–Pitaevskii equation can smooth a Thomas–Fermi approximation for a condensate edge while retaining this factorization. Describing quantum depletion of a Bose condensate additionally requires noncondensate mode occupations and their contribution to the one-body density matrix.