Dynamics and thermalization of the nuclear spin bath in the single-molecule magnet<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Mn</mml:mi><mml:mn>12</mml:mn></mml:msub><mml:mtext>−</mml:mtext><mml:mi>ac</mml:mi></mml:mrow></mml:math>: Test for the theory of spin tunneling

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. Most of the attention is usually given to the dynamics of the ``central spin'' (i.e., the qubit), while little is known about the spin bath itself. Here, we present a detailed study of the dynamics of the nuclear spin bath in the ${\mathrm{Mn}}_{12}\text{\ensuremath{-}}\mathrm{ac}$ single-molecule magnet, probed by NMR experiments down to very low temperatures $(T\ensuremath{\simeq}20\phantom{\rule{0.3em}{0ex}}\mathrm{mK})$. The results are critically analyzed in the framework of the Prokof'ev-Stamp theory of nuclear-spin-mediated quantum tunneling. We find that the longitudinal relaxation rate of the $^{55}\mathrm{Mn}$ nuclei in ${\mathrm{Mn}}_{12}\text{\ensuremath{-}}\mathrm{ac}$ becomes roughly $T$ independent below $T\ensuremath{\simeq}0.8\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ 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 $T&lt;0.8\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, and can be explained qualitatively and quantitatively by the dipolar coupling between like nuclei in neighboring molecules. This intercluster nuclear spin diffusion mechanism is an essential ingredient for the global relaxation of the nuclear spin bath. We also show that the isotopic substitution of $^{1}\mathrm{H}$ by $^{2}\mathrm{H}$ 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---where only $T$-independent quantum tunneling fluctuations are present---the nuclear spins remain in thermal equilibrium with the lattice phonons, and we investigate the time scale 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. This calls for an extension of the spin-bath theory to include the effect of spin-phonon couplings in the nuclear-spin-mediated tunneling process.

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