BEC order parameter Ψ(r,t): macroscopic occupation of the ground state

Bogoliubov / Penrose-Onsager criterion for Bose-Einstein condensation: a one-body density matrix ρ_1(r, r') = ⟨Ψ̂†(r')Ψ̂(r)⟩ has a single eigenvalue of order N — the condensate fraction N₀/N — with an extended macroscopic eigenfunction. …