Creation of Dirac Monopoles in Spinor Bose-Einstein Condensates

We demonstrate theoretically that, by using external magnetic fields, one can imprint pointlike topological defects on the spin texture of a dilute Bose-Einstein condensate. The symmetries of the condensate order parameter render this topological defect to be accompanied with a vortex filament corresponding to the Dirac string of a magnetic monopole. The vorticity in the condensate coincides with the magnetic field of a magnetic monopole, providing an ideal analogue to the monopole studied by Dirac.

Figure 1

Possible configurations for the spin texture. Both configurations are characterized by the same charge $\mathcal{Q}=1$, but only (a) can be imprinted using the quadrupole fields. The hedgehog texture (b) describes the vorticity ${\mathbf{\Omega }}_s$ corresponding to the spin texture in (a). In both cases, the vector fields are symmetric with respect to rotations about the axis on which the singularity filament (zigzag line) lies.

Figure 2

Particle density at different stages of monopole creation. The monopole is created by ramping the bias field down in a period $t_0=50/{\omega }_r$. Blue (dark) color denotes densities from $\varrho =3.8\times {10}^{-5}N/a_r^3$ to $\varrho =1.8\times {10}^{-4}N/a_r^3$ and green (light) color densities from $\varrho =1.4\times {10}^{-3}N/a_r^3$ to $\varrho =1.5\times {10}^{-3}N/a_r^3$. The Dirac string is identified as the density depletion that propagates through the condensate.

Figure 3

Vorticity corresponding to the monopole. Vorticity ${\mathbf{\Omega }}_s$ is computed numerically after the bias field is ramped from $B_0(0)=1\mu \mathrm{T}$ to $B_0(t_0)=0$, and it is shown in the units of $h/m{a}_r$. (a) $t_0=50/{\omega }_r$ and the bias field is parallel to the $z$ axis. (b) $t_0=40/{\omega }_r$ and the bias field is parallel to the vector $3/2\widehat{\mathit{x}}+\widehat{\mathit{y}}+14/3\widehat{\mathit{z}}$. The magnitude of the vorticity is denoted by color, and the map is linear between the minimum and maximum values. For clarity, only the relevant parts of the vector field ${\mathbf{\Omega }}_s$ are shown.

Figure 4

Unwinding of the monopole defect in the spin texture $\mathcal{S}$ (a)–(d) and in the vorticity ${\mathbf{\Omega }}_S$ (e)–(h). Arrows pointing upwards (downwards) from the plane are denoted by ⊙ ($\otimes$). Along the dashed line in (b) and (c), the spin points either upwards or downwards from the plane. The unwinding in ${\mathbf{\Omega }}_S$ is shown in a plane perpendicular to the $z$ axis, and it is qualitatively independent of the $z$ coordinate. High-resolution figures from the simulation corresponding to these schematic illustrations are available in Ref. 26.