Two-dimensional solitons in dipolar Bose-Einstein condensates with spin-orbit coupling

We report families of two-dimensional (2D) composite solitons in spinor dipolar Bose-Einstein condensates, with two localized components linearly mixed by the spin-orbit coupling (SOC), and the intrinsic nonlinearity represented by the dipole-dipole interaction (DDI) between atomic magnetic moments polarized in plane by an external magnetic field. Recently, stable solitons were predicted in the form of semivortices (composites built of coupled fundamental and vortical components) in the 2D system combining the SOC and contact attractive interactions. Replacing the latter by the anisotropic long-range DDI, we demonstrate that, for a fixed norm of the soliton, the system supports a continuous family of stable spatially asymmetric vortex solitons (AVSs), parameterized by an offset of the pivot of the vortical component relative to its fundamental counterpart. The offset is limited by a certain maximum value, while the energy of the AVS practically does not depend on the offset. At small values of the norm, the vortex solitons are subject to a weak oscillatory instability. In the present system, with the Galilean invariance broken by the SOC, the composite solitons are set in motion by a kick the strength of which exceeds a certain depinning value. The kicked solitons feature a negative effective mass, drifting along a spiral trajectory opposite to the direction of the kick. A critical angular velocity, up to which the semivortices may follow rotation of the polarizing magnetic field, is found too.

Figure 1

(a) The stable SVS with total norm $N=0.15$. (a1,a2) Intensity distributions of the fundamental and vortical components, $|{\varphi }_{+}{(x,y)|}^2$ and $|{\psi }_{-}{(x,y)|}^2$, respectively. (a3) The phase structure of the vortical component. (a4) Direct real-time simulations corroborating the stability of this SVS (the evolution of the vortical component is displayed in cross section $y=0$). The evolution time in this panel exceeds 100 characteristic diffraction times of the soliton. (b) An unstable SVS with $N=0.10$. (b1,b2) The same as in (a1,a2), for the unstable SVS. (b3,b4) Real-time simulations show the oscillatory instability of both components, in cross section $y=0$, by means of the density contour plots. The resulting breathers keeps the vorticity and symmetries of the original SVS.

Figure 2

(a1,a2) Horizontal and vertical sizes [defined per Eq. (14)] vs $N$ for the two SVS components. (b1) Anisotropies of both components, ${\eta }_{\pm }$, vs $N$ [defined as per Eq. (16)]. (b2) The norm share of the vortical component of the SVS, $F_{-}$, vs $N$ [defined as per Eq. (17)]. (c1,c2) The total energy of the SVS, and the energy of the spin-orbit coupling, $E_{\mathrm{SOC}}$, vs $N$. (d1) The chemical potential of the SVS, $\mu$, vs $N$. In the yellow area, stationary SVSs are subject to the oscillatory instability, transforming into breathers which keep the same structure as the SVSs. (d2) The critical rotation speed, ${\omega }_{\mathrm{cr}}$, vs $N$.

Figure 3

(a,b,c) Trajectories of the fundamental c.m. of the soliton kicked by $\mathbf{k}\equiv (k_x,k_y)=(0.05,0)$ (blue cycle) and $(0,0.05)$ (red cycle) [panel (a)], $(0.25,0)$ (blue spiral) and $(0,0.25)$ (red spiral) [panel (b)], and $(0.5,0)$ (blue lacy curve) and $(0,0.5)$ (red lacy curve) [panel (c)], respectively. In all the cases, the soliton's norm is $N=0.15$. In these panels, the evolution time is $t=40$. Panel (d) displays the mean velocity of the spiral motion of the soliton vs $k_x$ for $N=0.15$ and $0.2$, respectively. ${\overline{V}}_x<0$ means that the soliton moves against the direction of the applied kick.

Figure 4

(a1–a4) Typical examples of the density profile in the fundamental (a1,a2) and vortical (a3,a4) component, ${\varphi }_{+}$ and ${\varphi }_{-}$, of a stable AVS, for the initial offset of the vortex's pivot with respect to the fundamental component $(X_{\mathrm{ps}}^{\mathbf{in}},Y_{\mathrm{ps}}^{\mathbf{in}})=(0.01,0)$ and $(0,0.01)$, respectively. The origin of the coordinate system is placed at the final position of the pivot. The total norm of the asymmetric semivortex soliton is $N=0.15$. (a5,a6) The phase structure of the vortical components (${\psi }_{-}$) from panels (a3) and (a4), respectively. The output values of the pivot's offset of these two AVSs are $(X_{\mathrm{ps}}^{\mathrm{out}},Y_{\mathrm{ps}}^{\mathrm{out}})=(0.04,0)$ and $(0,0.065)$, respectively. (b1) The energy of the AVS vs $X_{\mathrm{ps}}^{\mathrm{in}}$, showing the degeneracy of the family. (b2) $X_{\mathrm{ps}}^{\mathrm{out}}$ vs $X_{\mathrm{ps}}^{\mathrm{in}}$, the saturation taking place at the largest output offset $X_{\mathrm{ps}}^{\max }=0.47$, for the AVS with $N=0.15$. (b3) The largest output offset of the vortex's pivot along the $x$ axis, $X_{\mathrm{ps}}^{\mathrm{out}}$, vs $N$.

Figure 5

(a1,a2) Trajectories of AVSs, with $\left(X_{\mathrm{ps}}^{\mathrm{out}},Y_{\mathrm{ps}}^{\mathrm{out}}\right)=(0,0)$ (blue curves), $(0.12,0)$ (green curves), and $(0.26,0)$ (red curves), produced by the application of the kick $\mathbf{k}\equiv (k_x,k_y)=(+0.25,0)$ (a1) and $(+0.5,0)$ (a2), respectively. (b1,b2) Trajectories of the AVSs with $\left(X_{\mathrm{ps}}^{\mathrm{out}},Y_{\mathrm{ps}}^{\mathrm{out}}\right)=(0.12,0)$ (b1) and $(0.26,0)$ (b2), kicked by $\mathbf{k}\equiv (k_x,k_y)=(+0.25,0)$ (blue curves), $(-0.25,0)$ (red curves), and $(0,+0.25)$ (green curves). (c1,c2) Trajectories of the AVSs with $\left(X_{\mathrm{ps}}^{\mathrm{out}},Y_{\mathrm{ps}}^{\mathrm{out}}\right)=(0.12,0)$ (c1) and $(0.26,0)$ (c2) kicked by $\mathbf{k}\equiv (k_x,k_y)=(+0.5,0)$ (blue curves), $(-0.5,0)$ (red curves), and $(0,+0.5)$ (green curves).

Figure 6

The stable rotation of the vortical component of the semivortex solitons driven by the rotating magnetic field. (a1–a3) The symmetric semivortex. (b,c) Asymmetric semivortices,with $(X_{\mathrm{ps}}^{\mathrm{out}},Y_{\mathrm{ps}}^{\mathrm{out}})=(0.04,0)$ and $(X_{\mathrm{ps}}^{\mathrm{out}},Y_{\mathrm{ps}}^{\mathrm{out}})=(0,0.065)$, respectively. The rotation speed is $\omega =0.45\pi$, and the total norm of each semivortex soliton is $N=0.15$. The respective evolution of the fundamental component is not shown, as it is less conspicuous.