Anomalous Spatial Diffusion and Multifractality in Optical Lattices

The transport of cold atoms in shallow optical lattices is characterized by slow, nonstationary momentum relaxation. We develop a projector operator method able to derive, in this case, a generalized Smoluchowski equation for the position variable. We show that this explicitly non-Markovian equation can be written as a systematic expansion involving higher-order derivatives. We use the latter to compute arbitrary moments of the spatial distribution and analyze their multifractal properties.

Figure 1

Second moment of the position distribution, Eq. (16), for three different values of the parameter $\alpha$ (black solid). The colored dots are the results of Langevin simulations. Parameters are $D_0=1$ and $t^{*}=5000$.

Figure 2

Fourth moment of the position distribution, Eq. (17), for three different values of the parameter $\alpha$ (black solid). The colored dots are the results of Langevin simulations. Parameters are $D_0=1$ and $t^{*}=5000$.

Figure 3

Exponent $\xi (q)$ of the moments $⟨|x{|}^q(\tau )⟩\propto {\tau }^{\xi (q)}$ as a function of the order $q$, for different values of $\alpha$. The solid line corresponds to normal Gaussian diffusion ($\xi (q)=q/2$); whereas, the dash-dotted line represents quasiballistic expansion ($\xi (q)=q$). The solid symbols are obtained by fitting the long time behavior of Langevin simulations. The multifractality of the process is clearly visible, as $\xi (q)$ is not a linear function of $q$. It interpolates between normal diffusion for lower-order and quasiballistic motion for higher-order moments. Same parameters as in Fig. 1.