  We analyze the dephasing dynamics of an impurity coupled to an anharmonic
environment. We show that a strong anharmonicity produces two different effects
depending on the environment temperature: for high temperatures, the system
suffers a strong dephasing, while for low temperatures there is a strong
information back-flow (as measured by the Breuer-Laine-Piilo (BLP)
non-Markovianity measure). Both dephasing and back-flow are particularly
significant when the anharmonic potential allows environment states very close
to the dissociation limit. In contrast, the information back-flow is suppressed
when assuming the environment to be Gaussian. In this regard, we find that the
Gaussian approximation is particularly poor at low temperatures.
