Quantum dynamics of bosons in a two-ring ladder: dynamical algebra, vortex-like excitations and currents

We study the quantum dynamics of the Bose-Hubbard model on a ladder formed by two rings coupled by tunneling effect. By implementing the Bogoliubov approximation scheme, we prove that, despite the presence of the inter-ring coupling term, the Hamiltonian decouples in many independent sub-Hamiltonians ${\widehat{H}}_k$ associated to momentum-mode pairs $\pm k$. Each sub-Hamiltonian ${\widehat{H}}_k$ is then shown to be part of a specific dynamical algebra. The properties of the latter allow us to perform the diagonalization process, to find energy spectrum, the conserved quantities of the model, and to derive the time evolution of important physical observables. We then apply this solution scheme to the simplest possible closed ladder, the double trimer. After observing that the excitations of the system are weakly-populated vortices, we explore the corresponding dynamics by varying the initial conditions and the model parameters. Finally, we show that the inter-ring tunneling determines a spectral collapse when approaching the border of the dynamical-stability region.