Spin-Orbit Coupled Spinor Bose-Einstein Condensates

An effective spin-orbit coupling can be generated in a cold atom system by engineering atom-light interactions. In this Letter we study spin-$1/2$ and spin-1 Bose-Einstein condensates with Rashba spin-orbit coupling, and find that the condensate wave function will develop nontrivial structures. From numerical simulation we have identified two different phases. In one phase the ground state is a single plane wave, and often we find the system splits into domains and an array of vortices plays the role of a domain wall. In this phase, time-reversal symmetry is broken. In the other phase the condensate wave function is a standing wave, and it forms a spin stripe. The transition between them is driven by interactions between bosons. We also provide an analytical understanding of these results and determine the transition point between the two phases.

Figure 1

An example of real (a) and imaginary (b) part of unstable branch of excitation spectrum with SO coupling.

Figure 2

Numerical results for the spin-$1/2$ case. a1 and a2 show the phase of condensate wave function of both spin-up (a1) and down (a2) component increase from $-\pi$ (grey regime) to $\pi$ (dark regime) periodically in the PW regime; b1 and b2 show the density of both the spin-up (b1) and down (b2) component oscillate periodically in the SW regime. $\gamma =c_2/c_0=0.4$ for a1 and a2, and $\gamma =0.1$ for b1 and b2.

Figure 3

Numerical results for spin-1 case. a1–a3 are density of 1, 0, and $-1$ component in the SW regime; and b1–b3 are phase of and b3 are 1, 0, and $-1$ component in the PW regime. $\gamma =c_2/c_0=0.2$ for a1–a3, and $\gamma =-0.2$ for b1–b3.

Figure 4

(a) Density of the up (or down) component with two domains in the PW phase. (b) A schematic of the vortex array as a domain wall between two regimes with opposite wave vector. (c)–(d) Consider array of vortices separated by distance $l$ located in $x$ axes, ${\partial }_x\theta$ (c) and ${\partial }_y\theta$ (d) (in unit of $1/l$) at position ($x$, $y$) as a function of $x/l$ is shown for $|y|=l$ (red dashed line), $|y|=1.5l$ (blue dotted line) and $|y|=2l$ (black solid line). $\theta$ is the phase of condensate wave function.