  We investigate the quantum dynamics of a harmonically trapped particle (e.g.
an ion) that is immersed in a Bose--Einstein condensate. The ultracold
environment acts as a refrigerator, and thus, the influence on the motion of
the ion is dissipative. We study the fully coupled quantum dynamics of particle
and Bose gas in a linearized regime, treating the quasi-particle excitations of
the gas as a (non-Markovian) environment for the particle dynamics. The density
operator of the latter follows a known non-Markovian master equation with a
highly non-trivial bath correlation function that we determine and study in
detail. The corresponding damping rate and frequency shift of the particle
oscillations can be read off. We are able to identify a Quantum Landau
criterion for harmonically trapped particles in a superfluid environment: for
frequencies $\omega$ well below the chemical potential, the damping rate is
strongly suppressed by a power law $\omega^4$. This criterion can be seen as
emerging from the classical Landau criterion involving a critical velocity
combined with Heisenberg's uncertainty principle for the localized wave packet
of the quantum particle. Furthermore, due to the finite size of the Bose gas,
after some time we observe memory effects and thus non-Markovian dynamics of
the quantum oscillator.
