Anomalous localization in a kicked quasicrystal

Quantum transport can distinguish between dynamical phases of matter. For instance, ballistic propagation characterizes the absence of disorder, whereas in many-body localized phases, particles do not propagate for exponentially long times. Additional possibilities include states of matter exhibiting anomalous transport in which particles propagate with a non-trivial exponent. Here we report the experimental observation of anomalous transport across a broad range of the phase diagram of a kicked quasicrystal. The Hamiltonian of our system has been predicted to exhibit a rich phase diagram, including not only fully localized and fully delocalized phases but also an extended region comprising a nested pattern of localized, delocalized and multifractal states, which gives rise to anomalous transport. Our cold-atom realization is enabled by new Floquet engineering techniques, which expand the accessible phase diagram by five orders of magnitude. Mapping transport properties throughout the phase diagram, we observe disorder-driven re-entrant delocalization and sub-ballistic transport, and we present a theoretical explanation of these phenomena based on eigenstate multifractality.