Solvable model of a trapped mixture of Bose–Einstein condensates

Abstract

A mixture of two kinds of identical bosons held in a harmonic potential and interacting by harmonic particle–particle interactions is discussed. This is an exactly-solvable model of a mixture of two trapped Bose–Einstein condensates which allows us to examine analytically various properties. Generalizing the treatments in Cohen and Lee (1985) and Osadchii and Muraktanov (1991), closed form expressions for the mixture’s frequencies and ground-state energy and wave-function, and the lowest-order densities are obtained and analyzed for attractive and repulsive intra-species and inter-species particle–particle interactions. A particular mean-field solution of the corresponding Gross–Pitaevskii theory is also found analytically. This allows us to compare properties of the mixture at the exact, many-body and mean-field levels, both for finite systems and at the limit of an infinite number of particles. We discuss the renormalization of the mixture’s frequencies at the mean-field level. Mainly, we hereby prove that the exact ground-state energy per particle and lowest-order intra-species and inter-species densities per particle converge at the infinite-particle limit (when the products of the number of particles times the intra-species and inter-species interaction strengths are held fixed) to the results of the Gross–Pitaevskii theory for the mixture. Finally and on the other end, we use the mixture’s and each species’ center-of-mass operators to show that the Gross–Pitaevskii theory for mixtures is unable to describe the variance of many-particle operators in the mixture, even in the infinite-particle limit. The variances are computed both in position and momentum space and the respective uncertainty products compared and discussed. The role of the center-of-mass separability and, for generically trapped mixtures, inseparability is elucidated when contrasting the variance at the many-body and mean-field levels in a mixture. Our analytical results show that many-body correlations exist in a trapped mixture of Bose–Einstein condensates made of any number of particles. Implications are briefly discussed.