Fragmentation of a Trapped Multi-Species Bosonic Mixture

We consider a multi-species mixture of interacting bosons, (Formula presented.) bosons of mass (Formula presented.), (Formula presented.) bosons of mass (Formula presented.), and (Formula presented.) bosons of mass (Formula presented.), in a harmonic trap with frequency (Formula presented.). The corresponding intra-species interaction strengths are (Formula presented.), (Formula presented.), and (Formula presented.), and the inter-species interaction strengths are (Formula presented.), (Formula presented.), and (Formula presented.). When the shape of all interactions is harmonic, the system corresponds to the generic multi-species harmonic-interaction model, which is exactly solvable. We start by solving the many-particle Hamiltonian and concisely discussing the ground-state wavefunction and energy in explicit forms as functions of all parameters, the masses, numbers of particles, and the intra-species and inter-species interaction strengths. We then explicitly compute the reduced one-particle density matrices for all the species and diagonalize them, thus generalizing the treatment by the authors earlier. The respective eigenvalues determine the degree of fragmentation of each species. As an application, we focus on phenomena that do not arise in the corresponding single-species or two-species systems. For instance, we consider a mixture of two kinds of bosons in a bath made by a third kind, controlling the fragmentation of the former by coupling to the latter. Another example exploits the possibility of different connectivities (i.e., which species interacts with which species) in the mixture, and demonstrates how the fragmentation of species 3 can be manipulated by the interaction between species 1 and species 2, when species 3 and 1 do not interact with each other. We highlight the properties of fragmentation that only appear in the multi-species mixture. Further applications are briefly discussed.