  We study the dynamics of spin currents in the XX spin-1/2 ladder at finite
temperature. Within the framework of linear response theory, we numerically
calculate autocorrelation functions for quantum systems larger than what is
accessible with exact diagonalization using the concept of dynamical quantum
typicality. We show that spin Drude weights vanish exponentially fast with
increasing system size. As a main result, we unveil qualitatively different
dependencies of the spin diffusion coefficient on the rung-interaction
strength, resulting from a crossover from exponential to Gaussian dissipation
as the rung coupling increases. This behavior is also derived analytically. We
further discuss the relation of our results to experiments with cold atomic
gases.
