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

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.

revised version, to appear in Phys. Rev. C

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• Alpha cluster structures and monopole excitations in ${}^{13}$C
• Nonlocalized Clustering and Evolution of Cluster Structure in Nuclei
• Nuclear Alpha-Particle Condensates
• Trapped two-component Fermi gases with up to six particles: Energetics, structural properties, and molecular condensate fraction
• Finite-temperature dynamics of a single vortex in a Bose-Einstein condensate: Equilibrium precession and rotational symmetry breaking
• Collision dynamics and entanglement generation of two initially independent and indistinguishable boson pairs in one-dimensional harmonic confinement
• Dynamics of quantum vortices at finite temperature