  Conventional wisdom holds that macroscopic classical phenomena naturally
emerge from microscopic quantum laws. However, despite this mantra, building
direct connections between these two descriptions has remained an enduring
scientific challenge. In particular, it is difficult to quantitatively predict
the emergent "classical" properties of a system (e.g. diffusivity, viscosity,
compressibility) from a generic microscopic quantum Hamiltonian. Here, we
introduce a hybrid solid-state spin platform, where the underlying disordered,
dipolar quantum Hamiltonian gives rise to the emergence of unconventional spin
diffusion at nanometer length scales. In particular, the combination of
positional disorder and on-site random fields leads to diffusive dynamics that
are Fickian yet non-Gaussian. Finally, by tuning the underlying parameters
within the spin Hamiltonian via a combination of static and driven fields, we
demonstrate direct control over the emergent spin diffusion coefficient. Our
work opens the door to investigating hydrodynamics in many-body quantum spin
systems.
