Internal One-Particle Density Matrix for Bose-Einstein Condensates with Finite Number of Particles in a Harmonic Potential

TL;DR

This paper investigates how the internal one-particle density matrix in Bose-Einstein condensates depends on the choice of internal coordinates, proposing criteria to select the physically appropriate coordinate system, and finds that Jacobi coordinates satisfy these criteria.

Contribution

The paper introduces criteria for selecting internal coordinate systems in Bose-Einstein condensates and shows Jacobi coordinates uniquely satisfy these, resolving ambiguities in the internal density matrix.

Findings

Jacobi coordinates satisfy the proposed criteria for internal density matrices.

Different internal coordinate choices lead to different condensate descriptions.

The criteria ensure the internal density matrix matches the laboratory frame in the large N limit.

Abstract

Investigations on the internal one-particle density matrix in the case of Bose-Einstein condensates with a finite number ($N$) of particles in a harmonic potential are performed. We solve the eigenvalue problem of the Pethick-Pitaevskii-type internal density matrix and find a fragmented condensate. On the contrary the condensate Jacobi-type internal density matrix gives complete condensation into a single state. The internal one-particle density matrix is, therefore, shown to be different in general for different choices of the internal coordinate system. We propose two physically motivated criteria for the choice of the adequate coordinate systems which give us a unique answer for the internal one-particle density matrix. One criterion is that in the infinite particle number limit ($N=\infty$) the internal one-particle density matrix should have the same eigenvalues and eigenfunctions as those of the corresponding ideal Bose-Einstein condensate in the laboratory frame. The other criterion is that the coordinate of the internal one-particle density matrix should be orthogonal to the remaining $(N - 2)$ internal coordinates. This second criterion is shown to imply the first criterion. It is shown that the internal Jacobi coordinate system satisfies these two criteria while the internal coordinate system adopted by Pethick and Pitaevskii for the construction of the internal one-particle density matrix does not. It is demonstrated that these two criteria uniquely determine the internal one-particle density matrix which is identical to that calculated with the Jacobi coordinates. The relevance of this work concerning $\alpha$-particle condensates in nuclei, as well as bosonic atoms in traps, is pointed out.