Excited states of two-dimensional solitons supported by spin-orbit coupling and field-induced dipole-dipole repulsion

It was recently found that excited states of semivortex and mixed-mode solitons are unstable in spin-orbit-coupled Bose-Einstein condensates (BECs) with contact interactions. We demonstrate a possibility to stabilize such excited states in a setting based on repulsive dipole-dipole interactions induced by a polarizing field, oriented perpendicular to the plane in which the dipolar BEC is trapped. The strength of the field is assumed to grow in the radial direction $\sim r^4$. Excited states of semivortex solitons have vorticities $S$ and $S+1$ in their two components, each being an eigenstate of the angular momentum. They are fully stable up to $S=5$. The excited state of mixed-mode solitons feature interweaving necklace structures with opposite fractional values of the angular momentum in the two components. They are stable if they are built of dominant angular harmonics $\pm S$, with $S\le 4$. The characteristics and stability of these two types of previously unknown higher-order solitons are systematically analyzed. Their characteristic size is $\sim 10\mu \mathrm{m}$, with the number of atoms $\lesssim {10}^5$.

Figure 1

(a) The first and second rows: density patterns of the ${\varphi }_{+}$ and ${\varphi }_{-}$ components of the fundamental SVs and their excited states. The third row: the total-density patterns of the binary BEC, i.e., $|{\varphi }_{+}{|}^2+{\left|{\varphi }_{-}\right|}^2$. The fourth and fifth rows: phase patterns of ${\varphi }_{+}$ and ${\varphi }_{-}$, respectively. From left to right: $S_{+}=0,1,2,3$, and 4. (b) Similar results for the MMs and their excited states, with the same meaning of the rows as in (a). From left to right: $S_1=0,1,2,3$, and 4. The total norm of all the soliton modes displayed in the figure is $N=4$.

Figure 2

(a) The chemical potential $\mu$ and (b) energy $E$ for both excited states of SVs and MM as functions of the total norm $N$ (the coincidence of the curves for the excited states of SVs and MM is explained in the main text). Black curves with squares, red curves with circles, and blue curves with triangles correspond to $S_{+}=S_1=1,2$, and 3, respectively. (c) The total normalized orbital angular momentum values $\langle L\rangle$ of excited state of SVs versus $N$ for different values of $S_{+}$. (d) The normalized orbital angular momentum for component ${\varphi }_{+}$, i.e., $\langle L_{+}\rangle$, of excited states of MM versus $N$ for different values of $S_1$. (e) Values of $\langle L_{\pm }\rangle$ for excited states of MM ver- sus $S_1$ for $N=4$. (Note that $\langle L_{-}\rangle \equiv -\langle L_{+}\rangle$ for excited states of MM).

Figure 3

Long real-time evolution of the density pattern $|{\psi }_{+}{\left(t\right)|}^2$, with $3\%$ random noise added to the initial conditions in all panels here. (a) A stable excited state of SVs mode with $(N,S_{+})=(4,5)$. (b) An unstable excited state of SVs with $(N,S_{+})=(4,6)$ . (c) A stable excited state of MM with $(N,S_1)=(4,4)$. (d) An unstable excited state of MM with $(N,S_1)=(4,5)$. All the panels are shown as isosurfaces of $0.4\times |{\psi }_{+}^{max}{(t=0)|}^2$.