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An extension of Bogoliubov theory for a many-body system with a time scale hierarchy: the quantum mechanics of second Josephson oscillations

Adiabatic approximations are a powerful tool for simplifying nonlinear quantum dynamics, and are applicable whenever a system exhibits a hierarchy of time scales. Current interest in small nonlinear quantum systems, such as few-mode Bose-Hubbard models, warrants further development of adiabatic methods in the particular context of these models. Here we extend our recent work on a simple four-mode Bose-Hubbard model with two distinct dynamical time scales, in which we showed that among the perturbations around excited stationary states of the system is a slow collective excitation that is not present in the Bogoliubov spectrum. We characterized this mode as a resonant energy exchange with its frequency shifted by nonlinear effects, and referred to it as a second Josephson oscillation, in analogy with the second sound mode of liquid helium II. We now generalize our previous theory beyond the mean field regime, and construct a general Bogoliubov free quasiparticle theory that explicitly respects the system's adiabatic invariant as well the exact conservation of particles. We compare this theory to the numerically exact quantum energy spectrum with up to forty particles, and find good agreement over a significant range of parameter space.