Quasi-one-dimensional spin-orbit- and Rabi-coupled bright dipolar Bose-Einstein-condensate solitons

We study the formation of stable bright solitons in quasi-one-dimensional (quasi-1D) spin-orbit- (SO-) and Rabi-coupled two pseudospinor dipolar Bose-Einstein condensates (BECs) of ${}^{164}\mathrm{Dy}$ atoms in the presence of repulsive contact interactions. As a result of the combined attraction-repulsion effect of both interactions and the addition of SO and Rabi couplings, two kinds of ground states in the form of self-trapped bright solitons can be formed, a plane-wave soliton (PWS) and a stripe soliton (SS). These quasi-1D solitons cannot exist in a condensate with purely repulsive contact interactions and SO and Rabi couplings (no dipole). Neglecting the repulsive contact interactions, our findings also show the possibility of creating PWSs and SSs. When the strengths of the two interactions are close to each other, the SS develops an oscillatory instability indicating a possibility of a breather solution, eventually leading to its destruction. We also obtain a phase diagram showing regions where the solution is a PWS or SS.

Figure 1

Profiles of SO- and Rabi-coupled bright dipolar solitons for $N={10}^3$ atoms of ${}^{164}\mathrm{Dy}$ ($a_{dd}\approx 132.7a_0$ in units of Bohr radius $a_0$). We set the SO and Rabi couplings $\gamma =1.0$ and $\mathrm{\Gamma }=0.25$, respectively. From here on the couplings are used in dimensionless form. We consider different values of the repulsive atomic interaction (a) $a=0$, (b) $a=40a_0(\approx 2.1\mathrm{nm}),$ and (c) $a=77a_0(\approx 4.1\mathrm{nm})$. Inset: The real and imaginary parts of the wave function. Lengths are measured in units of $l_{\perp }=\sqrt{\hslash /m{\omega }_{\perp }}=1\mu \mathrm{m}$.

Figure 2

SO- and Rabi-coupled bright dipolar solitons. Here $N={10}^3$ atoms of ${}^{164}\mathrm{Dy}$ with $a=77a_0,\mathrm{\Gamma }=0.75$ and four different values of $\gamma$. The solid line depicts ${\left|\mathrm{\Phi }\left(z\right)\right|}^2$, and the dashed and dotted lines represent squared real and imaginary parts of the wave function, respectively. Lengths are measured in units of $l_{\perp }$.

Figure 3

The same as in Fig. 2, but for a fixed SOC $\gamma =1.0$, and four values of the Rabi coupling $\mathrm{\Gamma }$.

Figure 4

SO- and Rabi-coupled condensates of $N={10}^3$ atoms of ${}^{164}\mathrm{Dy}$ without contact interaction $(a=0)$. (a) $\mathrm{\Gamma }=1.0,\gamma =1.0,2.0$, and 2.5. (b) $\gamma =3.0,\mathrm{\Gamma }=2.0,3.0$, and 4.0. Lengths are measured in units of $l_{\perp }$.

Figure 5

RRMS size of the SO- and Rabi-coupled dipolar BEC solitons versus (a) the contact interaction $a$, with $a_{dd}=132.7a_0$, and (b) the dipolar length $a_{dd}$, with $a=30a_0(\approx 1.6\mathrm{nm})$. Here $N={10}^3$ atoms, $\mathrm{\Gamma }=3.0$, and $\gamma =1.5,2.5$. Lengths are measured in units of $l_{\perp }$.

Figure 6

$\mathrm{\Gamma }-\gamma$ phase diagram of SO- and Rabi-coupled bright dipolar solitons of $N={10}^3$ atoms of ${}^{164}\mathrm{Dy}$ with $a=50a_0(\approx 2.7\mathrm{nm})$.