Dynamics of Matter-Wave Solitons in a Ratchet Potential

We study the dynamics of bright solitons formed in a Bose-Einstein condensate with attractive atomic interactions perturbed by a weak bichromatic optical lattice potential. The lattice depth is a biperiodic function of time with a zero mean, which realizes a flashing ratchet for matter-wave solitons. We find that the average velocity of a soliton and the soliton current induced by the ratchet depend on the number of atoms in the soliton. As a consequence, soliton transport can be induced through scattering of different solitons. In the regime when matter-wave solitons are narrow compared to the lattice period the dynamics is well described by the effective Hamiltonian theory.

Figure 1

(a) Cumulative velocity of a soliton, $\overline{v}$, vs effective mass, $N$, calculated using Eq. (1) (solid line) and EPA (dashed); $\overline{v}=1$ corresponds to $3.5\mathrm{mm}/\mathrm{s}$. Parameters are: $V=0.3$, $\varphi =\pi /2$, $\omega =10$, $x_0(0)=0$. Inset: Cumulative velocity $\overline{v}$ vs driving frequency, $\omega$, for $N=4$ and $x_0(0)=0$ (solid line) and $x_0(0)=-\pi /2$ (dashed line). (b,c) Density plot of the mean field evolution, $|\Psi (x,t){|}^2$, shown for the marked points at $N=2$ and $N=4$ in (a).

Figure 2

(a) Average velocity of a soliton, $⟨\overline{v}⟩$, vs effective mass $N$, calculated using the GP model (solid line) and EPA (dashed). Parameters are: $V_0=0.3$, $\varphi =\pi /2$, $\omega =10$.

Figure 3

(a) Cumulative velocity of a soliton, $\overline{v}$, vs initial position, $x_0(0)$, for (a) $N=1$; (b) $N=2$; (c) $N=5$; calculated using the GP model (solid line) and EPA (dashed). (d) Poincaré section for the effective-particle model (5) in (c). Sections are taken at $t=2\pi l/\omega$ where $l$ is an integer. Parameters are: $V_0=0.3$, $\varphi =\pi /2$, $\omega =10$.

Figure 4

Density plot, $|\Psi (x,t){|}^2$, of the colliding solitons (a) with $N=4$ and $N=2.2$ initially located at $x_0(0)=0$ and $x_0(0)=4\pi$, respectively, and (b) with $N=4$ initially located at $x_0(0)=0$ and $x_0(0)=3\pi +1.2$. Parameters are: $V_0=0.3$. $\varphi =\pi /2$, $\omega =10$.