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

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.

Shortened to Brief report

References in corpus (9)

• Theory of Bose-Einstein condensation in trapped gases
• Alpha cluster condensation in 12C and 16O
• Analysis of previous microscopic calculations for second $0^{+}$ state in ${}^{12}$C in terms of 3-alpha particle Bose-condensed state
• Do attractive bosons condense?
• Alpha-particle condensation in 16O via a full four-body OCM calculation
• Single $\alpha$-particle orbits and Bose-Einstein condensation in ${}^{12}$C
• The alpha-particle in nuclear matter
• On the criterion for Bose-Einstein condensation for particles in traps
• Destruction of Bose-Einstein condensate by strong interactions

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• Concepts of alpha-particle condensation
• Comparing and contrasting nuclei and cold atomic gases
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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
• Internal One-Particle Density Matrix for Bose-Einstein Condensates with Finite Number of Particles in a Harmonic Potential
• Repulsive Four-Body Interactions of $\alpha$ Particles and Quasi-Stable Nuclear $\alpha$-Particle Condensates in Heavy Self-Conjugate Nuclei
• Relative and center-of-mass motion in the attractive Bose-Hubbard model
• Alpha-Particle Condensation in Nuclear Systems