Two-species boson mixture on a ring: A group-theoretic approach to the quantum dynamics of low-energy excitations

We investigate the weak excitations of a system made up of two condensates trapped in a Bose- Hubbard ring and coupled by an interspecies repulsive interaction. Our approach, based on the Bogoliubov approximation scheme, shows that one can reduce the problem Hamiltonian to the sum of sub-Hamiltonians (H) over cap k, each one associated to momentum modes +/- k. Each (H) over capk is then recognized to be an element of a dynamical algebra. This uncommon and remarkable property allows us to present a straightforward diagonalization scheme, to find constants of motion, to highlight the significant microscopic processes, and to compute their time evolution. The proposed solution scheme is applied to a simple but nontrivial closed circuit, the trimer. The dynamics of low- energy excitations, corresponding to weakly populated vortices, is investigated considering different choices of the initial conditions and the angular- momentum transfer between the two condensates is evidenced. Finally, the condition for which the spectral collapse and dynamical instability are observed is derived analytically.