Motion of solitons in one-dimensional spin-orbit-coupled Bose-Einstein condensates

Solitons play a fundamental role in the dynamics of nonlinear excitations. Here we explore the motion of solitons in one-dimensional Bose-Einstein condensates subjected to a spin-orbit coupling (SOC). We demonstrate that the spin dynamics of solitons is governed by a nonlinear Bloch equation. The spin dynamics affects the orbital motion of solitons leading to spin-orbit effects in the dynamics of macroscopic quantum objects (mean-field solitons). The latter perform oscillations with a frequency determined by the SOC, Raman coupling, and intrinsic nonlinearity. These findings reveal unique features of solitons affected by the SOC, which are confirmed by analytical considerations and numerical simulations of the underlying Gross-Pitaevskii equations.

Figure 1

The track of the spin density on the Bloch sphere (a), and the corresponding evolution of the spin components (b) and center-of-mass coordinate (c) of the soliton for the initially balanced state, with $\theta (t=0)=\pi /4$, and initial phase difference ${\phi }_{-}(t=0)=\pi /4$ [see Eq. (3)]. Other parameters are $\mathrm{\Omega }=0.5,\lambda =0.5\sqrt{\mathrm{\Omega }},\delta =0$, and $g=-10$. $z$ and $t$ are measured in units of $a_{\perp }$ and ${\omega }_{\perp }^{-1}$.

Figure 2

(a)–(c) The evolution of the density in the two components of the soliton produced by the GP simulations for the initially balanced state with $\theta (t=0)=\pi /4$ and ${\phi }_{-}(t=0)=\pi /4$. The parameters are $\mathrm{\Omega }=0.5,\lambda =0.5\sqrt{\mathrm{\Omega }},\delta =0$, and $g=-10$. The spin dynamics and center-of-mass motion of the soliton, generated by these simulations (circles), and by the variational approximation based on Eqs. (9)–(11) and (15) (solid lines), are depicted in (d) and (e). (f) and (g) display decay of the soliton in the case of a weaker atomic interaction, $g=-3$. $z$ and $t$ are measured in units of $a_{\perp }$ and ${\omega }_{\perp }^{-1}$.

Figure 3

(a)–(c) The same as in Fig. 2, but for the dark soliton, in the case of $g=10$. $z$ and $t$ are measured in units of $a_{\perp }$ and ${\omega }_{\perp }^{-1}$.

Figure 4

The results for bright-bright solitons in the trap with $\gamma =0.1$, the other parameters are the same as in Fig. 2. $z$ and $t$ are measured in units of $a_{\perp }$ and ${\omega }_{\perp }^{-1}$.