  The description of the tunneling of a macroscopic variable in the presence of
a bath of localized spins is a subject of great fundamental and practical
interest, and is relevant for many solid-state qubit designs. Instead of
focusing on the the "central spin" (as is most often done), here we present a
detailed study of the dynamics of the nuclear spin bath in the Mn12-ac
single-molecule magnet, probed by NMR experiments down to very low temperatures
(T = 20 mK). We find that the longitudinal relaxation rate of the 55Mn nuclei
in Mn12-ac becomes roughly T-independent below T = 0.8 K, and can be strongly
suppressed with a longitudinal magnetic field. This is consistent with the
nuclear relaxation being caused by quantum tunneling of the molecular spin, and
we attribute the tunneling fluctuations to the minority of fast-relaxing
molecules present in the sample. The transverse nuclear relaxation is also
T-independent for T < 0.8 K, and can be explained qualitatively and
quantitatively by the dipolar coupling between like nuclei in neighboring
molecules. We also show that the isotopic substitution of 1H by 2H leads to a
slower nuclear longitudinal relaxation, consistent with the decreased tunneling
probability of the molecular spin. Finally, we demonstrate that, even at the
lowest temperatures, the nuclear spins remain in thermal equilibrium with the
lattice phonons, and we investigate the timescale for their thermal
equilibration. After a review of the theory of macroscopic spin tunneling in
the presence of a spin bath, we argue that most of our experimental results are
consistent with that theory, but the thermalization of the nuclear spins is
not.
