Matter-wave solitons of collisionally inhomogeneous condensates

We investigate the dynamics of matter-wave solitons in the presence of a spatially varying atomic scattering length and nonlinearity. The dynamics of bright and dark solitary waves is studied using the corresponding Gross-Pitaevskii equation. The numerical results are shown to be in very good agreement with the predictions of the effective equations of motion derived by adiabatic perturbation theory. The spatially dependent nonlinearity is found to lead to a gravitational potential, as well as to a renormalization of the parabolic potential coefficient. This feature allows one to influence the motion of fundamental as well as higher-order solitons.

Figure 1

Top panels: the effective potential $V\left(x_0\right)$ as a function of the soliton center $x_0$ for a trap strength $\Omega =0.05$ for a fundamental bright soliton of amplitude $\eta \left(0\right)=1$, initially placed at the trap center $[x_0\left(0\right)=0]$. Different values of the gradient modify the character of the potential: in panel (a) it is purely attractive $(\delta =0,\Omega ∕2,\sqrt{2}\Omega )$, while in (b) it is either purely gravitational $(\delta =\sqrt{3}\Omega )$ or expulsive $(\delta =2\Omega )$. Bottom panels: evolution of the center of the bright soliton for the above cases: (c) for attractive effective potentials and (d) for gravitational or expulsive ones. The agreement between numerical results (solid and dashed lines) and the theoretical predictions (triangles, dots) is excellent.

Figure 2

(Color online) Spatiotemporal contour plots of the soliton densities for $\delta =\sqrt{2}\Omega$ (top panel) and $\delta =\sqrt{3}\Omega$ (bottom panel) up to a scale ${10}^{-4}$ (densities of ${10}^{-4}$ or greater are presented in dark red). In both cases $\Omega =0.05$. These contour plots correspond to an attractive or a gravitational effective potential respectively (see also Fig. 1). One can clearly observe the presence of the radiation emitted, which is detectable on a scale of ${10}^{-4}$.

Figure 3

(Color online) Spatiotemporal contour plot of the soliton density for $\Omega =0.075$, $\delta =\sqrt{3}\Omega$, and an optical lattice with $V_0=0.25$ and $\kappa =0.5$. One can clearly discern the presence of Bloch oscillations in the evolution of the density, whose period is in very good agreement with the corresponding theoretical prediction.

Figure 4

(Color online) Evolution of a double soliton with initial amplitude $A=2.5$ initially placed at the trap center $(x=0)$ with a strength $\Omega =0.1$ in the presence of a gradient $\delta =0.01$. Top panel: spatiotemporal contour plot of the density. Bottom panels: snapshots of the evolution of the density are shown for $t=0$ (top left panel), $t=10$ (top right panel), $t=30$ (bottom left panel), and $t=60$ (bottom right panel), covering almost one period of the oscillation. Dashed lines correspond to the trapping potential.

Figure 5

(Color online) (a) Two snapshots of the density of the dark soliton (at $t=0$ and $t=90$) on top of a Thomas-Fermi cloud. The chemical potential is $\mu =1$, the trap strength is $\Omega =0.05$, and the gradient is $\delta =0.01$. (b) The motion of the center of a dark soliton. Solid line and dots correspond to the numerical integration of the GP equation and analytical predictions [see Eq. (29)], respectively. (c) Spatiotemporal contour plot of the reduced condensate density—i.e., the actual density (density of the Thomas-Fermi cloud with the dark soliton) minus the Thomas-Fermi density—demonstrating the radiation, in the form of sound waves, emitted by the dark soliton. The radiation is detectable on a scale of ${10}^{-2}$.