Quasi-two-dimensional soliton in a self-repulsive spin-orbit-coupled dipolar binary condensate

TL;DR

This work investigates quasi-2D solitons in a spin-orbit-coupled binary Bose-Einstein condensate with both nondipolar and dipolar interactions. It develops a dimensionless quasi-2D Gross-Pitaevskii model including SOC and nonlocal dipolar terms, analyzes linear eigenfunctions to identify candidate modes, and then computes stationary solitons via imaginary-time propagation. For a nondipolar self-repulsive system, three degenerate solitons—\text{multi-ring}, stripe, and circularly-asymmetric—persist at weak to moderate SO coupling, with a square-lattice soliton appearing at intermediate SOC; in-plane dipolar interactions remove the square-lattice branch, leaving stripe and circularly-asymmetric solitons. In the dipolar case, the anisotropy introduced by polarization in the plane constrains the soliton landscape further, eliminating the square-lattice state and yielding orientation-dependent stripe solitons along with circularly-asymmetric solitons, while ensuring dynamical stability. These results illuminate how SO coupling and dipolar anisotropy sculpt 2D soliton structures and suggest feasible experiments in Dy-based or spinor BEC platforms to observe stripe and supersolid-like density patterns.

Abstract

We study the formation of solitons in a uniform quasi-two-dimensional (quasi-2D) spin-orbit (SO) coupled self-repulsive binary dipolar and nondipolar Bose-Einstein condensate (BEC) using the mean-field Gross-Pitaevskii equation. For a weak SO coupling, in a nondipolar BEC, one can have three types of degenerate solitons: a multi-ring soliton with intrinsic vorticity of angular momentum projection $+1$ or $-1$ in one component and 0 in the other, a circularly-asymmetric soliton and a stripe soliton with stripes in the density. For an intermediate SO couplings, the multi-ring soliton ceases to exist and there appears a square-lattice soliton with a spatially-periodic pattern in density on a square lattice, in addition to the degenerate circularly-asymmetric and stripe solitons. In the presence of a dipolar interaction, with the polarization direction aligned in the quasi-2D plane, only the degenerate circularly-asymmetric and stripe solitons appear.

Figures (6)

• Figure 1: Contour plot of density $n_j$ of a quasi-2D nondipolar ($d=0$) self-repulsive multi-ring $(\mp 1,0)$-type binary Rashba or Dresselhaus SO-coupled BEC soliton for components (a) $j=1$ ($n_1$), (b) $j=2$ ($n_2$) and (c) total density ($n$); the same of a stripe soliton for components (d) $j=1$ , (e) $j=2$ and (f) total density; and of a circularly-asymmetric soliton for (g) $j=1$, (h) $j=2$ and (i) total density. In this figure and in Figs. \ref{['fig3']}, \ref{['fig4']}, \ref{['fig5']} and \ref{['fig6']} the energy is given in the plots of total density in the last column. The parameters are $c_1=c_2=0.25, \gamma =0.5$. The plotted quantities in all figures of this paper are dimensionless.
• Figure 2: Contour plot of phase of wave-function components (a) $j=1$, and (b) $j=2$ of the quasi-2D Dresselhaus SO-coupled multi-ring soliton of Figs. \ref{['fig1']}(a)-(b); the same of wave-function components (c) $j=1$, and (d) $j=2$ of a Rashba SO-coupled soliton of Figs. \ref{['fig1']}(a)-(b); of wave-function components (e) $j=1$, and (f) $j=2$ of the Dresselhaus SO-coupled soliton of Figs. \ref{['fig1']}(g)-(h); of wave-function components (g) $j=1$, (h) $j=2$ of the Dresselhaus SO-coupled dipolar soliton of Figs. \ref{['fig4']}(g)-(h); of wave-function components (i) $j=1$, (j) $j=2$ of the Dresselhaus SO-coupled dipolar soliton of Figs. \ref{['fig5']}(a)-(b); of wave-function components (k) $j=1$, (l) $j=2$ of the Dresselhaus SO-coupled dipolar soliton of Figs. \ref{['fig5']}(d)-(e).
• Figure 3: Contour plot of dimensionless density $n_{2D}(x,y)$ of (a)-(c) a square-lattice soliton, (d)-(f) a stripe soliton, and (g)-(i) a circularly-asymmetric soliton in a self-repulsive nondipolar SO-coupled BEC. The left [middle] column shows the density of component $j=1 (n_1)$ [$j=2 (n_2)$], whereas the right column depicts the total density. The parameters of the model are $c_1=c_2=0.25$ and $\gamma=2$.
• Figure 4: Contour plot of dimensionless density $n_{2D}(x,y)$ of (a)-(c) a multi-ring soliton, (d)-(f) a stripe soliton with stripes along the $y$ axis, (g)-(i) a circularly-asymmetric soliton, and (j)-(l) a stripe soliton with stripes along the polarization $y$ axis. The left [middle] column shows the density of component $j=1 (n_1)$ [$j=2 (n_2)$], whereas the right column depicts the total density. The parameters of the model are $c_1=c_2=d=0.5,$ and $\gamma=0.5$.
• Figure 5: Contour plot of dimensionless density $n_{2D}(x,y)$ of (a)-(c) a stripe soliton with stripes along the polarization $y$ axis, (d)-(f) a circularly-asymmetric soliton, and (g)-(i) a stripe soliton with stripes along the $x$ axis. The left [middle] column shows the density of component $j=1 (n_1)$ [$j=2 (n_2)$], whereas the right column depicts the total density. The parameters of the model are $c_1=c_2=d=0.5, d=0.5$ and $\gamma=2$.