  Many advanced quantum techniques feature non-Gaussian dynamics, and the
ability to manipulate the system in that domain is the next-stage in many
experiments. One example of meaningful non-Gaussian dynamics is that of a
double-well potential. Here we study the dynamics of a levitated nanoparticle
undergoing the transition from an harmonic potential to a double-well in a
realistic setting, subjecting to both thermalisation and localisation. We
characterise the dynamics of the nanoparticle from a thermodynamic
point-of-view, investigating the dynamics with the Wehrl entropy production and
its rates. Furthermore, we investigate coupling regimes where the the quantum
effect and thermal effect are of the same magnitude, and look at suitable
squeezing of the initial state that provides the maximum coherence. The effects
and the competitions of the unitary and the dissipative parts onto the system
are demonstrated. We quantify the requirements to relate our results to a
bonafide experiment with the presence of the environment, and discuss the
experimental interpretations of our results in the end.
