Local management of the nonlinearity of Bose-Einstein condensates with pinched potentials

We present a proposal for the local control of the nonlinearity in quasi-one-dimensional Bose-Einstein condensates induced by a local pinching of the transverse confining potential. We investigate the scattering of bright matter-wave solitons through a pinched potential using numerical simulations of the full three-dimensional Gross-Pitaevskii equation and the corresponding effective one-dimensional model with spatially varying nonlinearity.

Figure 1

Representation of the spatial structure of a pinch of the confinement potential described by Eq. (12). The black surface represents the lower half of an isosurface of $V_{\mathrm{ext}}$ and illustrates the transition from a region with (a) ${\omega }_x={\omega }_y=\overline{\omega }$ to (b) ${\omega }_x=\overline{\omega }-\mathrm{\Delta }\omega$ and ${\omega }_y=\overline{\omega }+\mathrm{\Delta }\omega$ for $\mathrm{\Delta }\omega =0.4\overline{\omega }$. In both insets equipotential lines (dashed lines) as well the local density of the transverse wave function ${\chi }_0$ are represented.

Figure 2

Numerical simulations for the evolution of a bright soliton with initial amplitude $a_i=0.2$ and velocity $v_i=0.05$ scattered by a pinching of strength $\eta =0.4$ centered at $z=0$. (a) Timeframes of the transmission of the soliton. The black isosurface corresponds to an isosurface of $V_{\mathrm{ext}}$, and the yellow isosurface to the wave-function density ${\left|\mathrm{\Psi }\right|}^2$. (b) Spatiotemporal contour plots of the density ${\left|\phi \right|}^2$ obtained after integration of the transverse dependence. Animation of the simulation is included in the Supplemental Material [36].

Figure 3

Same as Fig. 2 for a soliton with initial amplitude $a_i=0.2$ and velocity $v_i=0.05$ scattered by a pinching of strength $\eta =0.5$ centered at $z=0$. Animation of the simulation is included in the Supplemental Material [36].

Figure 4

Numerical results of the 3D GPE (circles) and of the 1D effective model (crosses) for the outcome of the scattering process of a bright soliton with initial amplitude $a_i=0.2$ and velocity $v_i=0.05$, in comparison with the theoretical prediction (dashed lines). (a) Transmission coefficient in function of the pinch strength. Dependence of the critical value ${\eta }_c$ on the (b) initial amplitude and (c) initial velocity.