Fragmentation of identical and distinguishable bosons’ pairs and natural geminals of a trapped bosonic mixture

In a mixture of two kinds of identical bosons, there are two types of pairs: identical bosons’ pairs, of either species, and pairs of distinguishable bosons. In the present work, the fragmentation of pairs in a trapped mixture of Bose–Einstein condensates is investigated using a solvable model, the symmetric harmonic-interaction model for mixtures. The natural geminals for pairs made of identical or distinguishable bosons are explicitly contracted by diagonalizing the intra-species and inter-species reduced two-particle density matrices, respectively. Properties of pairs’ fragmentation in the mixture are discussed, the role of the mixture’s center-of-mass and relative center-of-mass coordinates is elucidated, and a generalization to higher-order reduced density matrices is made. As a complementary result, the exact Schmidt decomposition of the wave function of the bosonic mixture is constructed. The entanglement between the two species is governed by the coupling of their individual center-of-mass coordinates, and it does not vanish at the limit of an infinite number of particles where any finite-order intra-species and inter-species reduced density matrix per particle is 100% condensed. Implications are briefly discussed.