One-body density matrix ($G^{(1)}(\mathbf{r},{\mathbf{r}}^{\prime })=⟨a^{†}(\mathbf{r})a({\mathbf{r}}^{\prime })⟩$)

The one-body density matrixuses a creation operator and an annihilation operator to describe first-order spatial quantum coherence. Its diagonal is particle-number density and its trace is the expectation value of the particle number operator, $\mathrm{Tr}{G}^{(1)}=⟨N⟩$. Dividing by $⟨N⟩$ instead gives a unit-trace convention, so the normalization matters when interpreting its eigenvalues.

For bosons, an eigenvalue that remains proportional to particle number in the thermodynamic limit identifies Bose-Einstein condensation. This criterion extends occupation of a single-particle mode to interacting systems.^[ref]

In a homogeneous system of volume $V$, an occupied plane-wave mode contributesA constant long-distance limit therefore corresponds to a zero-momentum condensate; a moving condensate can retain a spatial phase. Finite local density or short-distance quantum coherence alone does not establish off-diagonal long-range order.

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.