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We study the quantum transport and the nonequilibrium vibrational states of aquantum dot embedded between a normal and a superconducting lead with thecharge on the quantum dot linearly coupled to a harmonic oscillator offrequency $\omega$. To the leading order in the charge-vibration interaction,we calculate the current and the nonequilibrium phonon occupation by theKeldsyh Green's function technique. We analyze the inelastic,vibration-assisted tunneling processes in the regime $\omega <\Delta$, with thesuperconducting energy gap $\Delta$, and for sharp resonant transmissionthrough the dot. When the energy $\varepsilon_0$ of the dot's level is close tothe Fermi energy $\mu$, i.e. $|\varepsilon_0-\mu|\ll \Delta$, inelastic Andreevreflections dominate up to voltage $eV\gtrsim\Delta$. The inelasticquasiparticle tunneling becomes the leading process when the dot's level isclose to the gap $|\varepsilon_0-\mu|\sim \Delta \pm \omega$. In both cases,the inelastic tunneling processes appear as sharp and prominent peaks - notbroadened by temperature - in the $I$-$V$ characteristic and pave the way forinelastic spectroscopy of vibrational modes even at temperatures $T \gg\omega$. We also found that inelastic Andreev reflections as well asquasiparticle tunneling induce a strong nonequilibrium state of the oscillator.In different ranges on the dot's level, we found that the current produces: (i)ground-state cooling of the oscillator with phonon occupation $n\ll 1$, (ii)accumulation of energy in the oscillator with $n\gg 1$ and (iii) a mechanicalinstability which is a precursor of self-sustained oscillations. We show thatground-state cooling is achieved simultaneously for several modes of differentfrequencies. Finally, we discuss how the nonequilibrium vibrational state canbe detected by the asymmetric behavior of the inelastic current peaks respectto the gate voltage.

Charge-vibration interaction effects in normal-superconductor quantum dots