Observation of anomalous diffusion and fractional self-similarity in one dimension

We experimentally study anomalous diffusion of ultra-cold atoms in a one dimensional polarization optical lattice. The atomic spatial distribution is recorded at different times and its dynamics and shape are analyzed. We find that the width of the cloud exhibits a power-law time dependence with an exponent that depends on the lattice depth. Moreover, the distribution exhibits fractional self-similarity with the same characteristic exponent. The self-similar shape of the distribution is found to be well-fitted by a L\'{e}vy distribution, but with a characteristic exponent that differs from the temporal one. Numerical simulations suggest that this may be due to correlations between the atoms' velocity and flight duration.