Beating dark-dark solitons and Zitterbewegung in spin-orbit-coupled Bose-Einstein condensates

We present families of beating dark-dark solitons in spin-orbit (SO) -coupled Bose-Einstein condensates. These families consist of solitons residing simultaneously in the two bands of the energy spectrum. The soliton components are characterized by two different spatial and temporal scales, which are identified by a multiscale expansion method. The solitons are “beating” ones, as they perform density oscillations. The characteristic frequency of the latter is relevant to Zitterbewegung (ZB) oscillations, which were recently observed in experiments with SO-coupled condensates [C. Qu et al., Phys. Rev. A 88, 021604(R) (2013); L. J. LeBlanc et al., New J. Phys. 15, 073011 (2013)]. We find that spin oscillations may occur, depending on the parity of each soliton branch, which consequently lead to ZB oscillations of the beating dark solitons. Analytical results are corroborated by numerical simulations, illustrating the robustness of the solitons.

Figure 1

The left panels show representative illustrations of the dispersion relation branches ${\omega }_{+}\left(k\right)$ (dashed line) and ${\omega }_{-}\left(k\right)$ (solid line), while the right panels show the respective group velocity dispersion coefficients ${\omega }_{+}^{\prime \prime }\left(k\right)$ (dashed line) and ${\omega }_{-}^{\prime \prime }\left(k\right)$ (solid line). The (red) circles denote the points of equal group velocity $v_g^{+}=v_g^{-}$ at the wave number $k_c$. The slopes of the (red) straight lines in the left panels are equal to the group velocity for each branch at $k=k_c$.

Figure 2

Top panel: profiles of soliton solutions of Eqs. (23) and (24), for ${\epsilon }^2{\mu }_{+}=0.26$; solid (blue) and dashed (red) lines correspond to ${\Phi }_{+}$ and ${\Phi }_{-}$, respectively. Bottom panel: the normalized difference between the number of atoms of dark and bright components, $\Delta N=\int \left[\right({\mu }_{+}-|{\Phi }_{+}{|}^2)-|{\Phi }_{-}{|}^2]dx/\int ({\mu }_{+}-|{\Phi }_{+}{|}^2)dx$, as a function of ${\mu }_{+}$ for the soliton branches (a)–(d), corresponding to functional forms indicated by Eqs. (25, 26, 27, 28). Parameter values are $\Omega /k_L^2=1.25$, $\delta =0$, $k=0$, and $\epsilon =0.1$. All depicted quantities are dimensionless.

Figure 3

Top row: initial condition corresponding to a soliton of branch (a) of Fig. 2. Second and third row: evolution of the densities $|{\Psi }_{\uparrow }{|}^2$ and $|{\Psi }_{\downarrow }{|}^2$, respectively (left), and a close-up showing in detail the oscillations (right). Parameters used are the same as in Fig. 2 with ${\epsilon }^2{\mu }_{+}=0.1$, ${\epsilon }^2{\mu }_{-}=0.095$. The lower three rows are similar to the above, but for a soliton of branch (b) with $k=-2$, and other parameters $\Omega /k_L^2=1.8$, ${\epsilon }^2{\mu }_{+}=0.2$, and ${\epsilon }^2{\mu }_{-}=0.195$. All depicted quantities are dimensionless.

Figure 4

The time dependence of the mean velocity $\langle v\left(t\right)\rangle$ normalized to the wave number $k_L$, for different soliton solutions. Solid (red) line at $\langle v\left(t\right)\rangle =0$ corresponds to a stationary soliton of branch (a). The larger amplitude solid (blue) and smaller amplitude solid (green) lines correspond to solitons of branch (b) at $k=0$ and $k=-2$, respectively. The dotted (black) lines correspond to the semianalytical result of Eq. (31), and the dashed (black) line indicates the shift from the center to the soliton with $k=-2$. Parameters used are $\Omega /k_L^2=1.8$ and ${\epsilon }^2{\mu }_{+}=0.26$. All depicted quantities are dimensionless.