  We study experimentally the far-from-equilibrium dynamics in ferromagnetic
Heisenberg quantum magnets realized with ultracold atoms in an optical lattice.
After controlled imprinting of a spin spiral pattern with adjustable wave
vector, we measure the decay of the initial spin correlations through
single-site resolved detection. On the experimentally accessible timescale of
several exchange times we find a profound dependence of the decay rate on the
wave vector. In one-dimensional systems we observe diffusion-like spin
transport with a dimensionless diffusion coefficient of 0.22(1). We show how
this behavior emerges from the microscopic properties of the closed quantum
system. In contrast to the one-dimensional case, our transport measurements for
two-dimensional Heisenberg systems indicate anomalous super-diffusion.
