Bright solitons in a two-dimensional spin-orbit-coupled dipolar Bose-Einstein condensate

We study a two-dimensional spin-orbit-coupled dipolar Bose-Einstein condensate with repulsive contact interactions by both the variational method and the imaginary-time evolution of the Gross-Pitaevskii equation. The dipoles are completely polarized along one direction in the two-dimensional plane to provide an effective attractive dipole-dipole interaction. We find two types of solitons as the ground states arising from such attractive dipole-dipole interactions: a plane-wave soliton with a spatially varying phase and a stripe soliton with a spatially oscillating density for each component. Both types of solitons possess smaller size and higher anisotropy than the soliton without spin-orbit coupling. Finally, we discuss the properties of moving solitons, which are nontrivial because of the violation of Galilean invariance.

Figure 1

Profiles of the density $n_{\uparrow ,\downarrow }={\left|{\mathrm{\Phi }}_{\uparrow ,\downarrow }\right|}^2$ of (a) spin $\uparrow$ and (b) spin $\downarrow$, (c) the total density $n_{\uparrow }+n_{\downarrow }$, the phase of (d) spin $\uparrow$ and (e) spin $\downarrow$ for (a1)–(e1) and (a2)–(e2) a plane-wave soliton with ${\gamma }_{12}=6$ and (a3)–(e3) and (a4)–(e4) a stripe soliton with ${\gamma }_{12}=10$. (a1)–(el) and (a3)–(e3) The solitons are obtained by the variational method. (a2)–(e2) and (a4)–(e4) The solitons are calculated by the imaginary-time evolution of the GPE (9). The dashed white line labels the $x=0$ line. Here $\gamma =8, {\gamma }_d/\gamma =0.67$, and $\alpha =2$.

Figure 2

Plot of (a) $a_x$ and (b) $a_y$ as a function of ${\gamma }_d/\gamma$ for the plane-wave solitons (dotted blue line), stripe solitons (dashed green line), and traditional solitons (solid red line) without spin-orbit coupling. The aspect ratio $\sqrt{a_y/a_x}$ of a soliton is displayed in the inset of (b). (c) Variational parameters with respect ${\gamma }_d/\gamma$, associated with $x_0$ (dash-dotted blue line) and $J_p$ (dashed blue line) for the plane-wave soliton and $J_x$ (solid green line) and $J_y$ (dotted green line) for the stripe soliton. (d) Total energy of the variational ansatz wave function and the wave function numerically obtained by the imaginary-time evolution of the GPE for both plane-wave and stripe solitons. The solid green line (stripe soliton) and dashed blue line (plane-wave soliton) correspond to the variational results, while the green circles and blue squares correspond to the Gross-Pitaevskii results. Here $\alpha =2, \gamma =8$, and ${\gamma }_{12}=6$ (${\gamma }_{12}=10$) for the plane-wave (stripe) soliton.

Figure 3

(a) Plot of $x_0$ (dash-dotted blue line) and $J_p$ (dashed blue line) for the plane-wave variational ansatz and $J_x$ (solid green line) and $J_y$ (dotted green line) for the stripe variational ansatz with respect to $a_x$ by the minimization of the energy $E_s^{\mathrm{PW}}$ and $E_s^{\mathrm{stripe}}$. (b) Minimum energy of $E_s^{\mathrm{PW}}$ (dotted blue line) and $E_s^{\mathrm{stripe}}$ (solid green line) as a function of $a_x$. Here $\alpha =2$.

Figure 4

Plot of (a) $a_x$ and (b) $a_y$ as a function of the spin-orbit coupling strength $\alpha$ for the plane-wave soliton (dotted blue line) and stripe soliton (dashed green line). The aspect ratio $\sqrt{a_y/a_x}$ of a soliton is plotted in the inset of (a). (c) Change of $x_0$ (dash-dotted blue line) and $J_p$ (dashed blue line) for the plane-wave soliton and $J_x$ (solid green line) and $J_y$ (dotted green line) for the stripe soliton with respect to $\alpha$. (d) Total energy plus ${\alpha }^2/2$ plotted as a function of $\alpha$. The solid green (for a stripe soliton) and dashed blue (for a plane-wave soliton) lines are calculated by the variational method, while the green circles (for a stripe soliton) and blue squares (for a plane-wave soliton) are numerically obtained by the imaginary-time evolution of the GPE. Here $\gamma =8, {\gamma }_d/\gamma =0.67$, and ${\gamma }_{12}=6$ (${\gamma }_{12}=10$) for the plane-wave (stripe) soliton.

Figure 5

Imbalance $I$ and width $l_x$ of spin $\uparrow$ of the solitons with respect to the velocity along the (a) $y$ and (b) $x$ directions. The insets display the enlarged figure in a small velocity region. Density and phase profiles of four typical moving solitons for spin $\uparrow$ corresponding to the different velocities in (a) and (b) are plotted in (c)–(f) where the horizontal and vertical coordinates are $x^{\prime }$ and $y^{\prime }$ respectively. Here $\alpha =2, \gamma =8, {\gamma }_{12}=10$, and ${\gamma }_d/\gamma =1$, corresponding to a stripe soliton when stationary.