  We study the quantum transport and the nonequilibrium vibrational states of a
quantum dot embedded between a normal and a superconducting lead with the
charge on the quantum dot linearly coupled to a harmonic oscillator of
frequency $\omega$. To the leading order in the charge-vibration interaction,
we calculate the current and the nonequilibrium phonon occupation by the
Keldsyh Green's function technique. We analyze the inelastic,
vibration-assisted tunneling processes in the regime $\omega <\Delta$, with the
superconducting energy gap $\Delta$, and for sharp resonant transmission
through the dot. When the energy $\varepsilon_0$ of the dot's level is close to
the Fermi energy $\mu$, i.e. $|\varepsilon_0-\mu|\ll \Delta$, inelastic Andreev
reflections dominate up to voltage $eV\gtrsim\Delta$. The inelastic
quasiparticle tunneling becomes the leading process when the dot's level is
close to the gap $|\varepsilon_0-\mu|\sim \Delta \pm \omega$. In both cases,
the inelastic tunneling processes appear as sharp and prominent peaks - not
broadened by temperature - in the $I$-$V$ characteristic and pave the way for
inelastic spectroscopy of vibrational modes even at temperatures $T \gg
\omega$. We also found that inelastic Andreev reflections as well as
quasiparticle 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 mechanical
instability which is a precursor of self-sustained oscillations. We show that
ground-state cooling is achieved simultaneously for several modes of different
frequencies. Finally, we discuss how the nonequilibrium vibrational state can
be detected by the asymmetric behavior of the inelastic current peaks respect
to the gate voltage.
