Criterion for Bose-Einstein condensation in traps and self-bound systems

The internal one-particle density matrix is discussed for Bose-Einstein condensates with a finite number of particles in a harmonic trap. It is found that the outcome of the diagonalization of the single-particle density matrix depends on the choice of the internal coordinates: The Pethick-Pitaevskii (PP) type of internal density matrix, whose analytical eigenvalues and eigenfunctions are evaluated, yields a fragmented condensate, while the Jacobi type of internal density matrix leads to an ideal condensate. We give a criterion for the choice of the internal coordinates: In the macroscopic limit the internal density matrix should have the same eigenvalues and eigenfunctions as those of the corresponding ideal Bose-Einstein condensate in the laboratory frame, this being a very physical condition. One choice satisfying this boundary condition is given by the internal Jacobi coordinates, while the internal coordinates with respect to the center of mass of the PP density matrix do not satisfy this condition. Based on our criterion, a general definition of the internal one-particle density matrix is presented in a self-bound system consisting of interacting bosons.