Bose-Einstein condensates with localized spin-orbit coupling: Soliton complexes and spinor dynamics

Spin-orbit (SO) coupling can be introduced in a Bose-Einstein condensate (BEC) as a gauge potential acting only in a localized spatial domain. The effect of such a SO “defect” can be understood by transforming the system to the integrable vector model. The properties of the SO BEC change drastically if the SO defect is accompanied by the Zeeman splitting. In such a nonintegrable system, the SO defect qualitatively changes the character of soliton interactions and allows for formation of stable nearly scalar soliton complexes with almost all atoms concentrated in only one dark state. These solitons exist only if the number of particles exceeds a threshold value. We also report on the possibility of transmission and reflection of a soliton upon its scattering on the SO defect. Scattering strongly affects the pseudospin polarization and can induce pseudospin precession. The scattering can also result in almost complete atomic transfer between the dark states.

Figure 1

Families of monopole solitons for $w=1.6$ (a) and dipole solitons for $w=1.5$ (b). In all cases $a=1,\Omega =1$. Stable (unstable) families are shown in black (red). The circles correspond to solitons shown in Figs. 2, 2, 2, and 2.

Figure 2

Monopole modes from lower (a, b) and upper (c, d) branches at $\mu =-2,w=1.6$ shown in Fig. 1. Panels (a, c) correspond to $a=1$, while panels (b, d) correspond to $a=0.1$. Dipole modes from the lower (e, f) and upper (g, h) branches at $\mu =-2,w=1.5$ are shown in Fig. 1. Panels (e, g) correspond to $a=1$, while panels (f, h) correspond to $a=0.1$. Tripole solitons are from the lower branch with $\mu =-4,w=2,a=2$ (i), and $a=0.6$ (j). Quadrupole solitons are from the lower branch with $\mu =-4,w=4,a=4$ (k), and $a=1.3$ (l). In all the cases $\Omega =1$. Dashed lines show $\kappa \left(x\right)$ profiles.

Figure 3

Domains of stability (white) and instability (shaded) for the monopole (a, b) and dipole (c, d) solitons from the lower branches in Figs. 1 and 1, respectively. In (a, c) the defect width $w=1.6$, while in (b, d) the defect amplitude $a=1$. In all cases $\Omega =1$.

Figure 4

Splitting of stable modes into solitons after switching off SO coupling at $t=50$ (dashed line). The initial distributions correspond to the monopole at $\mu =-2.5,a=1,w=1.6$ (a); dipole at $\mu =-4.2,a=1,w=1.5$ (b); and tripole at $\mu =-4,a=1.5,w=3$ (c). In all cases $\Omega =1$ and the total evolution time is $t=200$.

Figure 5

Soliton interaction with the SOD in the presence of the Zeeman splitting $\Omega =0.099$ (a) and $\Omega =0.1$ (b). The parameters of the incident soliton and the SO defect are the same in both cases: $\eta =1,\alpha =\pi /2,v=0.4,\beta =0$, and $a=w=1$. The evolution is shown up to $t=240$ in the window $x\in [-24,24]$. The pseudospin components are shown in the window $S_k\in [-3,3]$.