L\'evy flights in laser cooling of nuclear spins

Interaction between an electronic spin and its surrounding nuclear spin environment is a major source of decoherence in most solid-state spin qubits. We develop a phenomenological model for nuclear spin diffusion in the presence of electronic dark states. As an quantitative example, we study the diffusion in $^{13}$C nuclear spin bath of an NV$^-$ impurity in diamond. We use this model to predict that the nuclear diffusion time-scales exhibit L\'evy statistics--- enabling nuclear spins to remain trapped in certain configurations for long times. We comment on observing such statistics by measuring photon scattering rates that are dependent on nuclear diffusion rates, leading to quantitative measurements of the non-equilibrium bath dynamics in such central-spin systems.