Phase separation in a spin-orbit-coupled Bose-Einstein condensate

We study a spin-orbit (SO)-coupled hyperfine spin-1 Bose-Einstein condensate (BEC) in a quasi-one-dimensional trap. For an SO-coupled BEC in a one-dimensional box, we show that in the absence of the Rabi term, any nonzero value of SO coupling will result in a phase separation among the components for a ferromagnetic BEC, like ${}^{87}\mathrm{Rb}$. On the other hand, SO coupling favors miscibility in a polar BEC, like ${}^{23}\mathrm{Na}$. In the presence of a harmonic trap, which favors miscibility, a ferromagnetic BEC phase separates, provided that the SO-coupling strength and number of atoms are greater than some critical value. The Rabi term favors miscibility irrespective of the nature of the spin interaction: ferromagnetic or polar.

Figure 1

(a, b) Analytical (anal.) and numerical (num.) densities ${\rho }_j\left(x\right)l_0$ of a condensate of ${}^{87}\mathrm{Rb}$ atoms with $c_0=885.72l_0$ and $c_2=-4.09l_0$ in a 1D box of length $40l_0$. The SO coupling $\gamma =0$ and $0.5$ for (a) and (b), respectively. (c) The same for a condensate of ${}^{23}\mathrm{Na}$ atoms with $c_0=241.28l_0$ and $c_2=7.76l_0$ in the presence of an arbitrary SO coupling. Both the densities and the spatial coordinates are in dimensionless units.

Figure 2

Ground-state structure of an ${}^{87}\mathrm{Rb}$ spinor BEC with $10000$ atoms with $\Omega =0$. The SO coupling $\gamma =0$, 0.25, 0.5, 1 for (a), (b), (c), and (d), respectively. All quantities in are dimensionless.

Figure 3

Ground-state structure of an ${}^{87}\mathrm{Rb}$ spinor BEC with $10000$ atoms and $\Omega =0,\gamma =1$. Both the density and the spatial coordinates are plotted in dimensionless units. The magnetization $\mathcal{M}=0.25,0.5,0.75$, and $0.95$ for (a), (b), (c), and (d), respectively.

Figure 4

Ground-state structure of an ${}^{87}\mathrm{Rb}$ spinor BEC with ${10}^4$ atoms with (a) $N=10000,\Omega =0.1,\gamma =1$; (b) $N=10000,\Omega =1,\gamma =1$; (c) $N=1000,\Omega =0.1,\gamma =1$; and (d) $N=1000,\Omega =1,\gamma =1$. Both the density and the spatial coordinates are plotted in dimensionless units. The magnetization $\mathcal{M}=0$ in all the cases.

Figure 5

Ground-state structure of a ${}^{23}\mathrm{Na}$ spinor BEC with (a) $N=10000,\Omega =0,\gamma =1$ and, also, $\gamma =0$; (b) $N=10000,\Omega =0.5,\gamma =1$; (c) $N=10000,\Omega =1,\gamma =1$; and (d) $N=10000,\Omega =1.5,\gamma =1$. Both the density and the spatial coordinates are plotted in dimensionless units. The magnetization $\mathcal{M}=0$ is 0 in all cases.