  Disorder, noise and interaction play a crucial role in the transport
properties of real systems, but they are typically hard to control and study
both theoretically and experimentally, especially in the quantum case. Here we
explore a paradigmatic problem, the diffusion of a wavepacket, by employing
ultra-cold atoms in a disordered lattice with controlled noise and tunable
interaction. The presence of disorder leads to Anderson localization, while
both interaction and noise tend to suppress localization and restore transport,
although with completely different mechanisms. When only noise or interaction
are present we observe a diffusion dynamics that can be explained by existing
microscopic models. When noise and interaction are combined, we observe instead
a complex anomalous diffusion. By combining experimental measurements with
numerical simulations, we show that such anomalous behavior can be modeled with
a generalized diffusion equation, in which the noise- and interaction-induced
diffusions enter in an additive manner. Our study reveals also a more complex
interplay between the two diffusion mechanisms in regimes of strong interaction
or narrowband noise.
