Observation of Anomalous Diffusion and Fractional Self-Similarity in One Dimension

Yoav Sagi, Miri Brook, Ido Almog, and Nir Davidson

Phys. Rev. Lett. 108, 093002 – Published 1 March 2012

Abstract

We experimentally study anomalous diffusion of ultracold 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 Lévy distribution, but with a characteristic exponent that differs from the temporal one. Numerical simulations suggest that this is due to long trapping times in the lattice and correlations between the atom’s velocity and flight duration.

Received 18 November 2011

DOI:https://doi.org/10.1103/PhysRevLett.108.093002

Authors & Affiliations

Yoav Sagi, Miri Brook, Ido Almog, and Nir Davidson

Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 108, Iss. 9 — 2 March 2012

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) The experimental setup as seen from above. The atoms are released from a small crossed dipole trap into a second far-off-resonance trap with a very elongated aspect ratio which confine their motion to a single direction (tube trap). This trap is superimposed with the two counterpropagating Sisyphus lattice beams which induce the diffusive motion. After a chosen time, the atomic density is recorded using absorption imaging technique. Each image gives the column two dimensional density, and we integrate over the radial directions to obtain the axial atomic distribution. (b) Atomic distributions after different diffusion times in a $4.8 E_{r}$ deep lattice. Each measurement is repeated 20 times and averaged to improve the signal to noise ratio.Reuse & Permissions
Figure 2
A log-log graph of the FWHM squared as a function of the time squared, for different lattice depths. To facilitate the comparison of the slopes, all graphs were shifted in the $y$ direction to cross at a single point. The linear behavior establishes the power-law time scaling of the width of the atomic distribution. The FWHM is found by fitting the atomic distributions to a general function from which we extract the width. In very shallow lattices (below $\sim 1.7 E_{r}$ ) we notice an excess density in the center of the distribution which is probably due to corrugation in the tube trap potential. To avoid systematic errors due to this effect, we exclude in this range of lattice depths the central part of the distribution when fitting and extracting the FWHM.Reuse & Permissions
Figure 3
The diffusion exponent $α$ as a function of the lattice depth. The exponent is extracted by three different methods; The (blue) squares are obtained by fitting the FWHM scaling to a power-law $FWHM \sim t^{1 / α}$ (the error bar represents a 95% confidence level). The (red) circles are obtained from the measure of the self-similar transformation. The (black) triangles are found by fitting the distributions taken after 30 ms with a Lévy $L_{α}$ function.Reuse & Permissions
Figure 4
The measure of the self-similarity, $m$ , as a function of the diffusion exponent, $α$ , for three different lattice depths. Each point is calculated from 13 distributions taken after 10–40 ms of diffusion and normalized such that their integral is unity. The inset shows an example of the rescaling transformation with $α = 1.25$ in a $4.8 E_{r}$ deep lattice.Reuse & Permissions
Figure 5
The atomic distributions after different diffusion times are fitted with a Lèvy $L_{α}$ function. The resulting exponent $α$ is plotted as a function of the diffusion time for three characteristic lattices. In the inset the distributions after 30 ms of diffusion and their corresponding fits are shown.Reuse & Permissions

Physical Review Letters