Nonlinear Resonant Transport of Bose-Einstein Condensates

The coherent flow of a Bose-Einstein condensate through a quantum dot in a magnetic waveguide is studied. By the numerical integration of the time-dependent Gross-Pitaevskii equation in the presence of a source term, we simulate the propagation process of the condensate through a double barrier potential in the waveguide. We find that resonant transport is suppressed in interaction-induced regimes of bistability, where multiple scattering states exist at the same chemical potential and the same incident current. We demonstrate, however, that a temporal control of the external potential can be used to circumvent this limitation and to obtain enhanced transmission near the resonance on experimentally realistic time scales.

Figure 1

External longitudinal potential $V$ in units of $\hslash {\omega }_{\perp }$. The dotted line shows the longitudinal atom density (in units of $a_{\perp }^{-1}$) of the scattering state associated with the fifth excited resonance, calculated at $\mu =1.127\hslash {\omega }_{\perp }$ and $j_t=1.6{\omega }_{\perp }$.

Figure 2

Drag exerted by the condensate onto the obstacle, plotted as a function of the chemical potential $\mu$ and the total current $j_t$ (in units of ${\omega }_{\perp }$) for $g=0$ (left panel) and $g=0.034\hslash {\omega }_{\perp }{a}_{\perp }$ (right panel). Light gray areas correspond to a low and dark gray areas to a high drag. In white areas, the integration of Eq. (3) leads to a diverging $A(x)$ (which indicates an inherently time-dependent behavior of the condensate). The location of the fifth and sixth excited resonance, where the drag vanishes, is marked by dashed lines.

Figure 3

Transmission spectrum obtained from the time-dependent propagation approach for $g=0.034\hslash {\omega }_{\perp }{a}_{\perp }$ and fixed incident current $j_i=1.6{\omega }_{\perp }$ (solid line) compared, in the upper panel, to the interaction-free spectrum (dotted line). The crosses show the results of a full three-dimensional calculation [25]. The dashed line in the lower panel shows the two other branches of the fifth resonance peak, which are not populated by the time-dependent propagation process.

Figure 4

Time evolution of the transmission coefficient $T$ (upper panel) during the ramping process of the external potential $V_0$ (lower panel) which shifts the effective chemical potential from $\mu =0.985$ to $1.125\hslash {\omega }_{\perp }$. As shown in the insets (with scaling as in Fig. 1), the wave function adiabatically evolves into a nearly resonant state with transmission close to unity and decays from there to the low-transmission scattering state within a time scale of the order of $\tau \sim 100{\omega }_{\perp }^{-1}$.