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We investigate the quantum dynamics of a harmonically trapped particle (e.g.an ion) that is immersed in a Bose--Einstein condensate. The ultracoldenvironment acts as a refrigerator, and thus, the influence on the motion ofthe ion is dissipative. We study the fully coupled quantum dynamics of particleand Bose gas in a linearized regime, treating the quasi-particle excitations ofthe gas as a (non-Markovian) environment for the particle dynamics. The densityoperator of the latter follows a known non-Markovian master equation with ahighly non-trivial bath correlation function that we determine and study indetail. The corresponding damping rate and frequency shift of the particleoscillations can be read off. We are able to identify a Quantum Landaucriterion for harmonically trapped particles in a superfluid environment: forfrequencies $\omega$ well below the chemical potential, the damping rate isstrongly suppressed by a power law $\omega^4$. This criterion can be seen asemerging from the classical Landau criterion involving a critical velocitycombined with Heisenberg's uncertainty principle for the localized wave packetof 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 ofthe quantum oscillator.

Cooling and frequency shift of an impurity in a ultracold Bose gas using an open system approach