Binary mixture in a 3-well ring: Characterization of amplification effects within the dynamical algebra so(2,3)

We investigate the quantum dynamics of a mixture formed by two interacting condensates in a ring geometry described by a Bose-Hubbard trimer. By implementing the Bogoliubov-approximation scheme we reduce the ring- trimer Hamiltonian to an effective quadratic form whose two-mode momentum operators constitute an algebra so(2,3). The latter, the dynamical algebra of the model, allows one to perform a complete diagonalization of the Hamiltonian, and to derive the exact time evolution of physical operators. While these aspects have been thoroughly studied in the stability regime of the ring trimer, the dynamical behavior in the instability regime has received little attention. To explore the latter, we violate the stability condition with a suitable change of the interspecies repulsion showing how this induces a spectral collapse, and drives the transition from purely-periodic oscillations of momentum-mode populations into an unstable dynamics featuring the populations’ amplification. We thoroughly analyze the mechanism behind this macroscopic effect demonstrating how it smoothly emerges from the structural metamorphosis of the Hamiltonian and of its algebraic properties. We compare this effect with the parametric amplification of optical devices.