  Recent progress in photonics has led to a renewed interest in time-varying
media that change on timescales comparable to the optical wave oscillation
time. However, these studies typically overlook the role of material dispersion
that will necessarily imply a delayed temporal response or, stated
alternatively, a memory effect. We investigate the influence of the medium
memory on a specific effect, i.e. the excitation of quantum vacuum radiation
due to the temporal modulation. We construct a framework which reduces the
problem to single-particle quantum mechanics, which we then use to study the
quantum vacuum radiation. We find that the delayed temporal response changes
the vacuum emission properties drastically: Frequencies mix, something
typically associated with nonlinear processes, despite the system being
completely linear. Indeed, this effect is related to the parametric resonances
of the light-matter system, and to the parametric driving of the system by
frequencies present locally in the drive but not in its spectrum.
