Solitons in Bose-Einstein Condensates with Helicoidal Spin-Orbit Coupling

We report on the existence and stability of freely moving solitons in a spatially inhomogeneous Bose-Einstein condensate with helicoidal spin-orbit (SO) coupling. In spite of the periodically varying parameters, the system allows for the existence of stable propagating solitons. Such states are found in the rotating frame, where the helicoidal SO coupling is reduced to a homogeneous one. In the absence of the Zeeman splitting, the coupled Gross-Pitaevskii equations describing localized states feature many properties of the integrable systems. In particular, four-parametric families of solitons can be obtained in the exact form. Such solitons interact elastically. Zeeman splitting still allows for the existence of two families of moving solitons, but makes collisions of solitons inelastic.

Figure 1

Dispersion curves ${\mu }_{\mathrm{l}}$ (blue lines) and ${\mu }_{\mathrm{u}}$ (red lines) for $\alpha =\kappa =1$ at $\mathrm{\Delta }=0$ (a) and $\mathrm{\Delta }=1$ (b); “$l$” and “$u$” stand for lower and upper branches. Arrows show bifurcation points for quiescent nonlinear modes.

Figure 2

(a) Families of the quiescent solitons at $\alpha =\mathrm{\Delta }=\kappa =1$. (b) Domains of existence and stability for quiescent “$+$” solitons on the plane ($\alpha$, $\mu$). Line with open circles shows upper cutoff for the chemical potential for “$+$” family. Line with red circles shows bottom of continuous spectrum (and, thus, the upper cutoff for the “$-$” family). Solitons are stable in the white domains and unstable in the gray domains.

Figure 3

Field modulus (solid lines) and current (dashed lines) in quiescent solitons belonging to “$+$” (a) and “$-$” (b) families at $\mu =-2.5$ and $\alpha =\mathrm{\Delta }=\kappa =1$. Black and red lines correspond to the first and second components.

Figure 4

Numbers of atoms and their distribution between the components $N_{1,2}$ in “$+$” (a) and “$-$” (b) families vs velocity $v$ at $\mu =-2.5$, $\alpha =1.5$, and $\mathrm{\Delta }=\kappa =1$. Domains of existence and stability for “$-$” (c) and “$+$” (d) solitons on the plane $(\alpha ,v)$. Solitons are stable in white areas and unstable in gray areas.

Figure 5

Collision of two solitons in the $(x,t)$ plane at $\mu =-2.5$, $\alpha =0.5$, and $\mathrm{\Delta }=\kappa =1$. Top and bottom rows show the atomic density distributions $s_0$ and distribution of the $z$ projection of the spin $s_3$, respectively. All left (right) solitons have initial velocity $v=+0.14$ ($v=-0.14$). The types of colliding solitons (“$+$” or “$-$”) are indicated on the plots. Transverse window $x\in [-30,30]$ is shown. Solitons evolve up to $t=150$. On the black-and-white panels lighter (darker) domains correspond to spin-up, $s_3>0$, (spin-down, $s_3<0$) states.