Effect of a potential step or impurity on the Bose-Einstein condensate mean field

The full set of stationary states of the mean field of a Bose-Einstein condensate in the presence of a potential step or pointlike impurity are presented in closed analytic form. The nonlinear Schrödinger equation in one dimension is taken as a model. The nonlinear analogs of the continuum of stationary scattering states, as well as evanescent waves, are discussed. The solutions include asymmetric soliton trains and other wave functions of intriguing form, such as a pair of dark solitons bound by an impurity.

Figure 1

Stationary solutions to the NLS with a potential step of the form $V\left(x\right)=V_0\theta \left(x\right)$. These solutions, which are the nonlinear analogs of the continuum of linear stationary scattering states, exhibit a large deviation from the traditional linear solutions. Shown are particular examples of (a) the density and (b) the phase for a repulsive interaction strength and (c) the density and (d) the phase for attractive interaction strength.

Figure 2

Stationary solutions to the NLS with a potential step for which the wave functions decay, which are the nonlinear analogs to evanescent waves. Shown are particular examples of the densities of nonlinear waves with (a) repulsive interaction strength, (b) attractive interaction strength, and (c) and (d) with an attractive interaction strength that decays on the lower side of the step.

Figure 3

Localized, symmetric solutions to the NLS with repulsive interaction strength in the presence of an impurity, $V\left(x\right)=V_0\delta \left(x\right)$. Shown are particular examples of (a) the density and (b) the phase of a dark soliton bound by a repulsive impurity, (c) the density and (d) the phase of a pair of dark solitons bound by an attractive impurity, and (e) the density and (f) the phase of a supercurrent deformed by an attractive impurity. Note that (a) and (b) may also be interpreted as deformations of a supercurrent.

Figure 4

Localized, symmetric solutions to the NLS with attractive interaction strength in the presence of an impurity. Shown are particular examples of (a) the density of a bright soliton, which is the ground-state solution to the NLS, deformed by a repulsive impurity and (b) the density of a bright soliton deformed by an attractive impurity.

Figure 5

Solutions to the NLS with an impurity that oscillates on one side of the potential. Shown are particular examples of (a) the density of a nonlinear wave with repulsive interaction strength and (b) the density of a nonlinear wave with attractive interaction strength. These waves have no analog with the solutions of the linear Schrödinger equation.

Figure 6

Nonsymmetric solutions to the NLS with an impurity. Shown are particular examples of (a) the density and (b) the phase of a nonlinear wave with repulsive interaction strength and (ac) the density and (d) the phase of a nonlinear wave with attractive interaction strength. These solutions are the nonlinear analogs to the continuum of linear stationary scattering states.