  We provide an experimental study of the relationship between the action of
different classical noises on the dephasing dynamics of a two-level system and
the non-Markovianity of the quantum dynamics. The two-level system is encoded
in the photonic polarization degrees of freedom and the action of the noise is
obtained via a spatial light modulator, thus allowing for an easy engineering
of different random environments. The quantum non-Markovianity of the dynamics
driven by classical Markovian and non-Markovian noise, both Gaussian and
non-Gaussian, is studied by means of the trace distance. Our study clearly
shows the different nature of the notion of non-Markovian classical process and
non-Markovian quantum dynamics.
