Self-trapping of Bose-Einstein condensates expanding into shallow optical lattices

We observe a sudden breakdown of the transport of a strongly repulsive Bose-Einstein condensate through a shallow optical lattice of finite width. We are able to attribute this behavior to the development of a self-trapped state by using accurate numerical methods and an analytical description in terms of nonlinear Bloch waves. The dependence of the breakdown on the lattice depth and the interaction strength is investigated. We show that it is possible to prohibit the self-trapping by applying a constant offset potential to the lattice region. Furthermore, we observe the disappearance of the self-trapped state after a finite time as a result of the revived expansion of the condensate through the lattice. This revived expansion is due to the finite width of the lattice.

Figure 1

(Color online) Schematic setup of the system. The BEC is initially located in the flat-bottom box reservoir A. The shutter to its right (dashed vertical line) can be removed instantaneously so that the atoms expand into the optical lattice $V_{\mathrm{opt}}\left(z\right)$ of size $L$. A wide potential-free region B serves as a second reservoir for the atoms. In our numerical calculations we used the dimensionless lengths $L_A=160\pi$, $L_B=326\pi$ and $L=10\pi$ (for an explanation of the units see text).

Figure 2

(Color online) Band structure for four sets of parameters $\mu$, $s$, and $n\beta$. The vertical line separates two sets with the same chemical potential (dashed line) but differing $s$ and $n\beta$. Note that each set is symmetric around $q=0$, hence we plot $\mid q\mid$. The offset $v$ is kept at zero. The other parameters are (a) $\mu =0.16$ $(\beta =79.58)$ with $s=0.13$, $n\beta =0.05$ (left part) and $s=0.25$, $n\beta =0.04$ (right part), (b) $\mu =0.795$ $(\beta =397.89)$ with $s=0.095$, $n\beta =0.329$ (left) and $s=0.127$, $n\beta =0.393$ (right). The left and right insets in (b) show a zoom of the loops near the left and right band edges, respectively.

Figure 3

(Color online) Upper panel: time-dependent particle number within the lattice. Lower panel: time-dependent currents at the beginning of the lattice (red solid line) and the last lattice site (blue dashed). The gray dashed horizontal line indicates the stationary current obtained as described in the text. Lower panel: The parameters are $\beta =397.89$, $v=0$ for all plots and (a),(c) $s=0.095$ and (b),(d) $s=0.127$. The vertical bar in (b) indicates the analytical result for the particle number difference $\mathrm{\Delta }N$ (see text).

Figure 4

(Color online) Stationary current for (a),(b) varying $s$ at $v=0$ and (b),(c) varying constant offset $v$ at fixed $s$. Parameters for (a) are $\beta =251.46$ (solid line), $\beta =318.31$ (dashed), and $\beta =397.89$ (dotted), for (b) $\beta =31.83$ (solid) and $\beta =79.58$ (dashed). Plot (c) is at $\beta =397.31$ for fixed $s=0.095$ (solid line) and $s=0.19$ (dashed), (d) is at $\beta =318.31$ with $s=0.095$ (solid) and $s=0.253$ (dashed). The values for the lattice amplitude $s$ used in (c),(d) are marked in (a) on the curves with the respective interaction strengths.

Figure 5

(Color online) Density plot of the BEC near the optical lattice. The lattice extends over $0\le z\le 10\pi$. In the upper panel light shades indicate high density, dark shades low density. For a typical rubidium BEC such as the one used in [13] the time scale of these plots is around $32\mathrm{ms}$, which is experimentally accessible. The lower panel shows the profile of the BEC at the times indicated by the corresponding horizontal lines in the upper panel. The dotted line indicates the analytical result Eq. (10) for the corresponding nonlinear Bloch wave and the thin solid gray line indicates the position of the optical lattice. For both plots $s=0.127$, $v=0$ and (a) $\beta =79.58$, (b) $\beta =397.89$.