  We analyze the dynamics of a Taub cosmological model in the presence of a
massless minimally coupled scalar field and a cosmological constant, in the
limit when both the Universe volume and the scalar field live in a
quasi-classical approximation. By other words, we study the dynamics of a
quantum small anisotropy evolving on a de Sitter background and in the presence
of a kinetic term of the inflaton field. We demonstrate that the quantum
anisotropy exponentially decays during the Universe expansion, approaching a
finite and small value. This result suggests that the quantum isotropization of
the Universe during a de Sitter phase is much weaker than the corresponding
classical evolution, favouring the survival of certain degree of anisotropy to
the de Sitter phase. Finally we analyze the case when also the scalar field is
considered as quantum variable, by showing how its variance naturally spreads
because of no potential term significantly affects its dynamics. This behaviour
results to be different from the anisotropy which is subjected to the potential
coming out from the spatial curvature.
