Solvable model of a generic trapped mixture of interacting bosons: Reduced density matrices and proof of Bose-Einstein condensation

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Abstract

A mixture of two kinds of identical bosons, species 1 with N 1 bosons of mass m 1 and species 2 with N 2 bosons of mass m 2, held in a harmonic potential of frequency ω and interacting by harmonic intra-species and inter-species particle-particle interactions of strengths , , and is discussed. This is an exactly-solvable model of a generic mixture of trapped interacting bosons which allows one to investigate and determine analytically properties of interest. To start, closed form expressions for the frequencies, ground-state energy, and wave-function of the mixture are obtained and briefly analyzed as a function of the masses, numbers of particles, and strengths and signs of interactions. To prove Bose-Einstein condensation of the mixture three steps are needed. First, we integrate the all-particle density matrix, employing a four-parameter matrix-recurrence relations, down to the lowest-order intra-species and inter-species reduced density matrices of the mixture. Second, the coupled Gross-Pitaevskii (mean-field) equations of the mixture are solved analytically. Third, we analyze the mixture's reduced density matrices in the limit of an infinite number of particles of both species 1 and 2 (when the interaction parameters, i.e. the products of the number of particles times the intra-species and inter-species interaction strengths, are held fixed) and prove that: (i) both species 1 and 2 are 100% condensed; (ii) the inter-species reduced density matrix per particle is separable and given by the product of the intra-species reduced density matrices per particle; and (iii) the mixture's energy per particle, and reduced density matrices and densities per particle all coincide with the Gross-Pitaevskii quantities. Finally, when the infinite-particle limit is taken with respect to, say, species 1 only (with interaction parameters held fixed) we prove that: (iv) only species 1 is 100% condensed and its reduced density matrix and density per particle, as well as the mixture's energy per particle, coincide with the Gross-Pitaevskii quantities of species 1 alone; and (v) the inter-species reduced density matrix per particle is nonetheless separable and given by the product of the intra-species reduced density matrices per particle. The results are compared and discussed with respect to the recent work by Bouvrie et al (2014 Eur. Phys. J. D 68 346) who found for attractive mixtures vanishing bipartite entanglement between the two species and between a single particle (of either kind) and the remaining particles in the mixture. Implications are briefly discussed.

Original languageEnglish
Article number295002
Number of pages24
JournalJournal of Physics A: Mathematical and Theoretical
Volume50
Issue number29
DOIs
StatePublished - 29 Jun 2017

Bibliographical note

Funding Information:
This research was supported by the Israel Science Foundation (Grant No. 600/15).

Publisher Copyright:
© 2017 IOP Publishing Ltd.

Keywords

  • Bose-Einstein condensation
  • Bosonic mixtures
  • reduced density matrices
  • solvable model

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modeling and Simulation
  • Mathematical Physics
  • Physics and Astronomy (all)

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