Scattering of matter waves in spatially inhomogeneous environments

We study scattering of quasi-one-dimensional matter waves at an interface of two spatial domains, one with repulsive and one with attractive interatomic interactions. It is shown that the incidence of a Gaussian wave packet from the repulsive to the attractive region gives rise to generation of a soliton train. More specifically, the number of emergent solitons can be controlled, e.g., by the variation of the amplitude or the width of the incoming wave packet. Furthermore, we study the reflectivity of a soliton incident from the attractive region to the repulsive one. We find the reflection coefficient numerically and employ analytical methods, which treat the soliton as a particle (for moderate and large amplitudes) or a quasilinear wave packet (for small amplitudes), to determine the critical soliton momentum (as a function of the soliton amplitude) for which total reflection is observed.

Figure 1

Motion of a Gaussian wave packet from the region with repulsive interactions to the region with attractive interactions. The top and middle panels show, respectively, the density profiles of the wave function at $t=0$ and $t=200$ [solid (blue) lines]; the normalized scattering length profile for $W=0.01$ is also depicted [dashed (red) line]. The bottom panel represents a contour plot showing the evolution of the density; the middle and bottom panels clearly show the creation of a bright soliton train. The parameter values are $U_0=1$, $x_0=20$, $l=10$, $K=1.5$, and $a_2/a_1=0.95$.

Figure 2

The top and bottom panels show, respectively, the number of solitons $N_s$ observed at $t=200$ as a function of the initial amplitude $U_0$ (for fixed $l=10$) or as a function of the initial wave packet width $l$ (for fixed $U_0=1$) of the Gaussian wave packet. The other parameter values are $W=0.01$, $a_1=1$, $a_2=0.95$, $K=1.5$, and $x_0=20$.

Figure 3

Contour plots showing the evolution of the density of a bright soliton of initial velocity $k=0.7$, scattered at the interface between the attractive and repulsive regions, at $x=0$ [depicted by the dashed (white) line]. From top to bottom, the amplitude of the soliton is $\eta =0.2$, $0.6$, and 1. The top (bottom) panel corresponds to total transmission (reflection); the middle panel shows partial reflection.

Figure 4

Reflection coefficient $R$ as a function of the initial soliton momentum $k$, for various values of large soliton amplitudes $\eta$. The inset shows cases corresponding to weak solitons.

Figure 5

Sketch of the effective potential $V_{\mathrm{eff}}$ (blue line) as a function of $x_0$. Shown also is the soliton, initially located at $x_0\left(0\right)$ [left (green) curve], far from the scattering length interface and in close proximity to the interface [right (red) curve], where the location of its center is $x_0\left(\right|\Delta x\left|\right)$.

Figure 6

Contour plot showing the space-time evolution of the motion of the soliton center. The bright soliton is initially placed at $x_0\left(0\right)=-10$, while its amplitude and momentum are $\eta =0.4$ and $k=0.1$, respectively. The dashed (black) curve represents the analytical result of Eq. (12), while the horizontal (white) line depicts $x=0$.

Figure 7

Critical momentum $k_{\mathrm{cr}}$, below which total reflection occurs, as a function of the soliton amplitude $\eta$. The (red) dots depict the numerical result, while the left (blue) and right (green) straight lines correspond to the analytical predictions of Eqs. (19) and (15), respectively. The parameter values are $a_1=1$, $a_2=0.95$, and $W=0.01$.