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

The internal one-particle density matrix is discussed for Bose-Einstein condensates with finite number of particles in a harmonic trap. The outcome of the digonalization of the density matrix depends on the choice of the internal coordinates: The Pethick-Pitaevskii-type internal density matrix, whose analytical eigenvalues and eigenfunctions are evaluated, yields a fragmented condensate, while the Jacobi-type 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 eigenvalues and eigenfunctions of an ideal Bose-Einstein condensate, this being a very physical condition for cases where the system is also an ideal Bose condensation in the laboratory frame. One choice fulfilling this boundary condition is given by the internal Jacobi coordinates, while the internal coordinates with respect to the center of mass do not satisfy the 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.