Solvable Model of a Generic Trapped Mixture of Interacting Bosons: Many-Body and Mean-Field Properties

A solvable model of a generic trapped bosonic mixture, N ₁ bosons of mass m ₁ and N ₂ bosons of mass m ₂ trapped in an harmonic potential of frequency ω and interacting by harmonic inter-particle interactions of strengths λ ₁, λ ₂, and λ ₁₂, is discussed. It has recently been shown for the ground state [J. Phys. A 50, 295002 (2017)] that in the infinite-particle limit, when the interaction parameters λ ₁(N ₁ - 1), λ ₂(N ₂ - 1), λ ₁₂ N ₁, λ ₁₂ N ₂ are held fixed, each of the species is 100% condensed and its density per particle as well as the total energy per particle are given by the solution of the coupled Gross-Pitaevskii equations of the mixture. In the present work we investigate properties of the trapped generic mixture at the infinite-particle limit, and find differences between the many-body and mean-field descriptions of the mixture, despite each species being 100%. We compute analytically and analyze, both for the mixture and for each species, the center-of-mass position and momentum variances, their uncertainty product, the angular-momentum variance, as well as the overlap of the exact and Gross-Pitaevskii wavefunctions of the mixture. The results obtained in this work can be considered as a step forward in characterizing how important are many-body effects in a fully condensed trapped bosonic mixture at the infinite-particle limit.