Already quite small quantum systems like the four-mode Bose-Hubbard model can be considered as a paradigm for mesoscopic quantum systems in thermal contact if it provides an intrinsic separation of time scales and thus adiabatic methods are suitable. In our previous work we could show by applying a series of Holstein-Primakoff transformations that in addition to coherent particle exchange there exists a further slow collective excitation corresponding to a resonant energy exchange mode not predicted by linear Bogoliubov theory. The frequency of this mode is sensitive to interactions among Bogoliubov quasi-particles and may be referred to as a second Josephson oscillation, in analogy to the second sound mode of liquid Helium II and might be interpreted as a toy model for heat exchange.
We now construct a generalized Bogoliubov free quasiparticle theory that explicitly respects the system's adiabatic invariant and the exact conservation of particles simultaneously which is able to capture this second Josephson mode. We compare this theory to the numerically exact quantum energy spectrum and find good agreement over a significant range of parameter space. Furthermore we explicitly identify these excitations in dynamical simulations. We directly compare classical mean field dynamics to the exact full quantum many-particle dynamics and show good agreement over a large range of the system parameters.
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