Second Josephson excitations beyond mean field as a toy model for thermal pressure: exact quantum dynamics and the quantum phase model

A simple four-mode Bose-Hubbard model with intrinsic time-scale separation can be considered as a paradigm for mesoscopic quantum systems in thermal contact. In our previous work, we showed that in addition to coherent particle exchange, a novel slow collective excitation can be identified by a series of Holstein-Primakoff transformations. This resonant energy exchange mode is not predicted by the linear Bogoliubov theory, and its frequency is sensitive to interactions among Bogoliubov quasi-particles; it may be referred to as a second Josephson oscillation, in analogy to the second sound mode of liquid helium II. In this paper, we will explore this system beyond the Gross-Pitaevskii mean-field regime. We directly compare the classical mean-field dynamics to the exact full quantum many-particle dynamics and show good agreement over a wide range of system parameters. The second Josephson frequency becomes imaginary for stronger interactions however, indicating dynamical instability of the symmetric state. By means of a generalized quantum phase model for the full four-mode system, we then show that, in this regime, high-energy Bogoliubov quasiparticles tend to accumulate in one pair of sites, while the actual particles preferentially occupy the opposite pair. We interpret this as a simple model of thermal pressure.