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

TL;DR

This paper establishes a criterion for selecting internal coordinates in Bose-Einstein condensates, demonstrating that the choice affects whether the condensate appears fragmented or ideal, with implications for understanding self-bound systems.

Contribution

It introduces a physical criterion for choosing internal coordinates in BECs, favoring Jacobi coordinates to correctly identify ideal condensation in finite and self-bound systems.

Findings

Jacobi coordinates lead to an ideal condensate in the macroscopic limit.

Pethick-Pitaevskii coordinates result in a fragmented condensate.

A general definition of the internal density matrix for self-bound systems is proposed.

Abstract

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.