Solvable Model of a Generic Trapped Mixture of Interacting Bosons: Many-Body and Mean-Field Properties at the Infinite-Particle Limit

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 $\omega$ and interacting by harmonic inter-particle interactions of strengths $\lambda_1$, $\lambda_2$, and $\lambda_{12}$, is discussed. It has recently been shown for the ground state [J. Phys. A {\bf 50}, 295002 (2017)] that in the infinite-particle limit, when the interaction parameters $\lambda_1(N_1-1)$, $\lambda_2(N_2-1)$, $\lambda_{12}N_1$, $\lambda_{12}N_2$ 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.