Bright solitons in spin-orbit-coupled Bose-Einstein condensates

We study bright solitons in a Bose-Einstein condensate with a spin-orbit coupling that has been realized experimentally. Both stationary bright solitons and moving bright solitons are found. The stationary bright solitons are the ground states and possess well-defined spin-parity, a symmetry involving both spatial and spin degrees of freedom; these solitons are real valued but not positive definite, and the number of their nodes depends on the strength of spin-orbit coupling. For the moving bright solitons, their shapes are found to change with velocity due to the lack of Galilean invariance in the system.

Figure 1

Stationary bright solitons at $\gamma =1.0$. The solid lines are numerical results, and the circles are from the variational method. In (a) and (b), $\alpha =1.0$. In (c) and (d), $\alpha =2.0$.

Figure 2

The relation between the number of soliton nodes $2\pi /J$ and the soliton width $S$. (a) Circles, squares, and stars are for $\gamma =1.0,1.5,2.0$, respectively. Here $\alpha$ increases from $0.5$ to $2.0$ (the arrow direction). (b) Circles, squares, and stars are for $\alpha =1.0,1.5,2.0$. Here $\gamma$ increases from $1.0$ to $4.0$ (the arrow direction). The open circle and square correspond to the bright solitons in Figs. 1a and 1b and Figs. 1c and 1d, respectively.

Figure 3

The spin polarization $\langle {\sigma }_z\rangle$ as a function of $\alpha$ and $\gamma$.

Figure 4

Moving bright soliton profiles ${\Phi }_v\left(x\right)$ from the numerical calculation (solid line) and the variational method (circles) with Eq. (9). $\gamma =1.0$. In (a) and (b), $\alpha =1.0$. In (c) and (d), $\alpha =2.0$.

Figure 5

Dynamical evolution of a bright soliton under the influence of a small linear potential $V\left(x\right)=\eta x$. $\alpha =1.7$, $\gamma =1.0$, $\eta =0.001$. For better comparison of shapes, the centers of the soliton densities have all been shifted to the zero point. The velocity of the soliton is labeled tot the right of (b).