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

The one-body density matrix contains first-order spatial coherence and mode occupations of a many-particle state. Its trace is the particle number in the convention displayed. A macroscopic eigenvalue identifies Bose-Einstein condensation; in a homogeneous condensate the long-distance matrix approaches a nonzero constant. A finite density or local coherence alone does not imply this long-distance limit.