Texture Control in a Pseudospin Bose-Einstein Condensate

We describe a wave function engineering approach to the formation of textures in nonrotated multicomponent Bose-Einstein condensates. With numerical simulations of a viable two-component condensate experiment, we demonstrate the formation of a ballistically expanding regular lattice texture, composed of half-quantum vortices and spin-2 textures. The formation is described by a linear interference process in which the geometry and phase of three initially separated wave packets provide deterministic control over the resulting lattice texture.

Figure 1

A planar lattice texture is created by the interference of a pseudospin-$\frac{1}{2}$ BEC initially separated into three pieces. The hexagonal lattice, visualized with Bloch vectors in this numerical simulation, has a motif containing three textures: one half-quantum vortex of each sign and a spin-2 texture.

Figure 2

${}^{87}\mathrm{Rb}$ hyperfine levels $|F=1,m_F=-1⟩\equiv |1⟩$ and $|F=2,m_F=+1⟩\equiv |2⟩$ are optically coupled with Gaussian and Laguerre-Gauss (LG) lasers, whose intensity profiles are shown. The ${\mathrm{LG}}_{p=0}^{\ell =1}$ phase is also shown winding from $-\pi$ (black) to $+\pi$ (white).

Figure 3

Numerical simulations of a two-component three-piece BEC generate vortex-antivortex lattices as the wave packets expand and interfere. For the initial phases shown, the lattices align as is evident in Fig. 4a.

Figure 4

(a) Lattices containing vortices (white circles) and antivortices (black circles) form in components ${\Psi }_1$ and ${\Psi }_2$. Three combinations form different textures in $\Psi \equiv ({\Psi }_1,{\Psi }_2)$. Gray circles indicate an aligned vortex and antivortex in the two components. (b) Wave packets with centers given by ${\mathbf{r}}_1\equiv \mathbf{0}$, ${\mathbf{r}}_2$, ${\mathbf{r}}_3$ have relative phases 0, ${\varphi }_2$, ${\varphi }_3$, respectively. The resulting vortex-antivortex lattice basis vectors are ${\mathbf{r}}_2^{\prime }$ and ${\mathbf{r}}_3^{\prime }$, with a motif containing a vortex and antivortex separated by ${\mathbf{r}}_0^{\prime }$. Fringe spacing $f$ is a convenient measure of unit cell size.

Figure 5

Three texture types within the motif: (i) a spin-2 defect, (ii) a ($+$) HQV, and (iii) a ($-$) HQV. (a) Vortex (v) and antivortex (av) cores are overlaid on a map of $|{\Psi }_1{\Psi }_2|$. (b) Bloch vectors are shown normalized to $({\Psi }_1^2+{\Psi }_2^2{)}^{1/2}$.