Nonlinear Self-Trapping of Matter Waves in Periodic Potentials

We report the first experimental observation of nonlinear self-trapping of Bose-condensed ${}^{87}\mathrm{R}\mathrm{b}$ atoms in a one-dimensional waveguide with a superimposed deep periodic potential . The trapping effect is confirmed directly by imaging the atomic spatial distribution. Increasing the nonlinearity we move the system from the diffusive regime, characterized by an expansion of the condensate, to the nonlinearity dominated self-trapping regime, where the initial expansion stops and the width remains finite. The data are in quantitative agreement with the solutions of the corresponding discrete nonlinear equation. Our results reveal that the effect of nonlinear self-trapping is of local nature, and is closely related to the macroscopic self-trapping phenomenon already predicted for double-well systems.

Figure 1

Observation of nonlinear self-trapping of Bose-condensed ${}^{87}\mathrm{R}\mathrm{b}$ atoms. The dynamics of the wave packet width along the periodic potential is shown for two different initial atom numbers. By increasing the number of atoms from $2000\pm 200$ (squares) to $5000\pm 600$ (circles), the repulsive atom-atom interaction leads to the stopping of the global expansion of the wave packet. The insets show that the wave packet remains almost Gaussian in the diffusive regime but develops steep edges in the self-trapping regime. These edges act as boundaries for the complex dynamics inside.

Figure 2

Comparison between theory and experiment for $s=10,7.6(5)\mu \mathrm{m}$ initial rms width and $5000\pm 600$ atoms. The upper graphs show the measured density distribution for different propagation times. During the initial expansion in the self-trapping regime the wave packet develops steep edges which act as stationary boundaries for the subsequent internal dynamics. The results of the numerical integration of Eq. (2) (depicted in the lower graphs) are in very good agreement. For $t=50\mathrm{m}\mathrm{s}$ a 1.5 mrad deviation of the waveguides’ horizontal orientation (consistent with the experimental uncertainty) is taken into account and reproduces the experimentally observed asymmetry (gray line).

Figure 3

A numerical investigation of the site-to-site tunneling dynamics. (a) The atomic distribution $N_j$ of the wave packet for $t=0$ and 50 ms. (b) The relative population difference $\Delta N_j$ time averaged over the expansion time indicates two regions with different dynamics. (c) The dynamics of $\Delta N_j$ and the phase difference $\Delta {\varphi }_j$ for the marked site oscillate around zero known as the zero-phase mode of the boson Josephson junction. (d) The dynamics in the edge region is characterized by long time periods where $|\Delta N_j|$ is close to 1 while at the same time $\Delta {\varphi }_j$ winds up very quickly (phase is plotted modulo $\pi$) known as “running phase self-trapping mode” in boson Josephson junctions. Thus the expansion of the wave packet is stopped due to the inhibited site-to-site tunneling at the edge of the wave packet.

Figure 4

Experimental investigation of the scaling behavior. The solid line shows the curve given by Eq. (4). Experimentally the parameter $\Lambda /{\Lambda }_c$ was varied by using three different periodic potential depths: $s=10.6(3)$ (stars), $11.1(3)$ (squares), and $11.5(3)$ (diamonds). For each potential depth, wave packets with different atom numbers and initial widths are prepared and the width for $t=50\mathrm{m}\mathrm{s}$ is determined. The experimental data show qualitatively the scaling behavior predicted by Eq. (4) and are in quantitative agreement with the results of the numerical integration of the DNL (dashed line). The inset depicts the nature of the scaling: increasing $\Lambda /{\Lambda }_c$ (by, e.g., increasing the atom number) leads to a faster trapping and thus to a smaller final width.