  We study diffusion in ratchet systems. As a particular experimental
realization we consider an asymmetric SQUID subjected to an external ac current
and a constant magnetic flux. We analyze mean-square displacement of the
Josephson phase and find that within selected parameter regimes it evolves in
three distinct stages: initially as superdiffusion, next as subdiffusion and
finally as normal diffusion in the asymptotic long-time limit. We show how
crossover times that separates these stages can be controlled by temperature
and an external magnetic flux. The first two stages can last many orders longer
than characteristic time scales of the system thus being comfortably detectable
experimentally. The origin of abnormal behavior is noticeable related to the
ratchet form of the potential revealing an entirely new mechanism of emergence
of anomalous diffusion. Moreover, a normal diffusion coefficient exhibits
non-monotonic dependence on temperature leading to an intriguing phenomenon of
thermal noise suppressed diffusion. The proposed setup for experimental
verification of our findings provides a new and promising testing ground for
investigating anomalies in diffusion phenomena.
