  The quantum phase-space dynamics driven by hyperbolic P\"oschl-Teller (PT)
potentials is investigated in the context of the Weyl-Wigner quantum mechanics.
The obtained Wigner functions for quantum superpositions of ground and
first-excited states exhibit some non-classical and non-linear patterns which
are theoretically tested and quantified according to a non-gaussian continuous
variable framework. It comprises the computation of quantifiers of
non-classicality for an anharmonic two-level system where non-Liouvillian
features are identified through the phase-space portrait of quantum
fluctuations. In particular, the associated non-gaussian profiles are
quantified by measures of {\em kurtosis} and {\em negative entropy}. As
expected from the PT {\em quasi}-harmonic profile, our results suggest that
quantum wells can work as an experimental platform that approaches the gaussian
behavior in the investigation of the interplay between classical and quantum
scenarios. Furthermore, it is also verified that the Wigner representation
admits the construction of a two-particle bipartite quantum system of
continuous variables, $A$ and $B$, identified by $\sum_{i,j=0,1}^{i\neq j}\vert
i_{_A}\rangle\otimes\vert j_{_B}\rangle$, which are shown to be separable under
gaussian and non-gaussian continuous variable criteria.
