  We show that the nuclear spin dynamics in the single-molecule magnet Mn12-ac
below 1 K is governed by quantum tunneling fluctuations of the cluster spins,
combined with intercluster nuclear spin diffusion. We also obtain the first
experimental proof that - surprisingly - even deep in the quantum regime the
nuclear spins remain in good thermal contact with the lattice phonons. We
propose a simple model for how T-independent tunneling fluctuations can relax
the nuclear polarization to the lattice, that may serve as a framework for more
sophisticated theories.
