Abstract
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.
Highlights
Condensation and fragmentation are basic and widely-studied concepts of Bose-Einstein condensate emanating from the properties of the reduced one-particle density matrix [1,2,3,4,5]
The bosons are said to be condensed if there is a single macroscopic eigenvalue of the reduced one-particle density matrix [6] and fragmented if there are two or more such macroscopic eigenvalues [7]
The analysis of the reduced two-particle density matrix would determine whether pairs of bosons are condensed or fragmented
Summary
Condensation and fragmentation are basic and widely-studied concepts of Bose-Einstein condensate emanating from the properties of the reduced one-particle density matrix [1,2,3,4,5]. The above discussion defines the goals of the present work which are: (i) To investigate fragmentation of pairs of identical bosons and establish fragmentation of pairs of distinguishable bosons in a mixture of Bose-Einstein condensates; (ii) To construct the respective natural geminals of the mixture, for identical pairs and for distinguishable pairs; (iii) To show that fragmentation of distinguishable bosons’ pairs in the mixture persists with higher-order inter-species reduced density matrices; (iv) To construct the Schmidt decomposition of the mixture’s wavefunction and discuss some of its properties at the limit of an infinite-number of particles where the mixture is 100% condensed; and (v) Achieving the first four goals analytically, using an exactly solvable model To this end we recruit the harmonic-interaction model for mixtures [71,72,73,74,75,76], or, more precisely here, a symmetric version of which [77]. Appendix A collects for comparison with the mixture the details of fragmentation of bosons and pairs in the single-species system
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