Observation of Subdiffusion in a Disordered Interacting System

We study the transport dynamics of matter-waves in the presence of disorder and nonlinearity. An atomic Bose-Einstein condensate that is localized in a quasiperiodic lattice in the absence of atom-atom interaction shows instead a slow expansion with a subdiffusive behavior when a controlled repulsive interaction is added. The measured features of the subdiffusion are compared to numerical simulations and a heuristic model. The observations confirm the nature of subdiffusion as interaction-assisted hopping between localized states and highlight a role of the spatial correlation of the disorder.

Figure 1

Time evolution of the width $\sigma$ for different initial interaction energies: $E_{\mathrm{int}}=0$ (squares), $E_{\mathrm{int}}=1.8J$ (triangles), and $E_{\mathrm{int}}=2.3J$ (circles). The continuous lines are the fit with Eq. (1). The dashed lines show the fitted asymptotic behavior, while the dash-dotted line shows the expected behavior for normal diffusion. The lattice parameters are $J/h=180\mathrm{Hz}$, $\Delta /J=4.9$.

Figure 2

Diffusion exponent $\alpha$ vs the initial interaction energy $E_{\mathrm{int}}$ in the experiment (triangles and squares) and simulations (circles). The experimental data are for $\Delta /J=5.3(4)$ and two different values of the tunneling: $J/h=180\mathrm{Hz}$ (triangles) and $J/h=300\mathrm{Hz}$ (squares). The vertical bars are the fitting error of Eq. (1) to the data, while the horizontal bars indicate the statistical error.

Figure 3

(a) Shape of an interacting sample after 10 s of evolution; the dashed curve is a Gaussian fit of the tails. (b) Time evolution of the kurtosis $\gamma$; the dashed line marks the kurtosis of a Gaussian distribution. The interaction energy is $E_{\mathrm{int}}\approx 1.8J$, all other parameters as in Fig. 2.

Figure 4

Effect of a finite radial temperature. The scattering length and mean temperature of the three data sets are: $a=100a_0$, $T_r=60\mathrm{nK}$ (circles); $a=400a_0$, $T_r=60\mathrm{nK}$ (triangles); $a=400a_0$, $T_r=200\mathrm{nK}$ (squares). The continuous lines are the fit with Eq. (1), while the dashed lines are the fit to the asymptotic behavior. The fitted exponents are: $\alpha =0.32(2)$ (circles); $\alpha =0.34(2)$ (triangles); $\alpha =0.40(5)$ (squares). The lattice parameters are $J/h=290\mathrm{Hz}$, $\Delta /J=3.9$.