Numerical study of the stability of skyrmions in Bose-Einstein condensates

We show that the stability of three-dimensional skyrmions in trapped Bose-Einstein condensates depends critically on scattering lengths, atom numbers, trap rotation, and trap anisotropy. In particular, for the ${}^{87}\mathrm{Rb}\mid F=1,m_f=-1⟩$, $\mid F=2,m_f=1⟩$ hyperfine states stability is sensitive to the scattering lengths at the 2% level, where the differences between them are crucial. In a cigar shaped trap, we find stable skyrmions with slightly more than $2\times {10}^6$ atoms, a number which scales with the inverse square root of the trap frequency. These can be stabilized against drift out of the trap by laser pinning.

Figure 1

(Color online) Three-dimensional density profiles for a skyrmion with $N_{\mathrm{tot}}=2\times {10}^6$, $N_r=0.6$ in the presence of a pinning potential [$V_0=6.5\hslash {\omega }_x$, $w_0=1.3{\sigma }_x$, see Eq. (7)] and in a rotating trap $(\Omega =0.06{\omega }_x)$. The anisotropy of the trap is ${\omega }_{\perp }∕{\omega }_z=1.3$ and the scattering lengths are the ${\overline{a}}_{ij}$. See text for definitions of symbols and for values of other parameters. The central (blue) torus is an isosurface of the line component. An isosurface of the ring component (red) is shown for $x<0$: on the $y\text{−}z$ plane between the isosurface sections the ring component density is indicated by a color map from red (lowest) to purple (highest). The circulation associated with each vortex is also indicated.

Figure 2

(Color online) Skyrmion stability diagram against cylindrically symmetric perturbations, in a spherical trap: (+ blue) stable, (⊗ red) unstable. The axes are the total atom number $N_{\mathrm{tot}}=N_1+N_2$ and relative line component fraction $N_r=N_2∕N_{\mathrm{tot}}$. (a) ${\overline{a}}_{ij}$ set of scattering lengths. For $N_r>0.6$ all tested cases were unstable. (b) $a_{ij}$ set of scattering lengths. For $N_r>0.65$ all tested cases were unstable.

Figure 3

(Color online) Mach number of ring component in the $y\text{−}z$ plane for a case near the stability boundary with $N_{\mathrm{tot}}=5\times {10}^6$ and $N_r=0.5$. The superimposed colormap shows the ring component density (blue, lowest density; red, highest density). The peaks are at the location of the ring singularity. Scattering lengths ${\overline{a}}_{ij}$.

Figure 4

Influence of changes in $N_r$ on the core $(N_{\mathrm{tot}}=5\times {10}^6)$: (Dashed) $N_r=0.47$, (solid) $N_r=0.5$, (dotted-dashed) $N_r=0.58$. Scattering lengths ${\overline{a}}_{ij}$. (a) Shape of the core. Shown is the cross section of an isosurface of the density of the line component $(n_2=2.5\times {10}^{-19}{\mathrm{m}}^{-3})$ with the $x\text{−}z$ plane. (b) Mach number of the ring component $M_1$ along the core for $x=0$, $y=0$.

Figure 5

Skyrmion core properties ($N_{\mathrm{tot}}=5\times {10}^6$ and $N_r=0.5$). Scattering lengths ${\overline{a}}_{ij}$. (a) Core width $d$ defined in the text. (b) Density of ring component. (c) Flow velocity $v_1$. (d) Mach number $M_1$ of the ring component along the core.

Figure 6

(Color online) Windows of stable rotation frequency for various total atom numbers; $N_r=0.5$ thoughout. (+ blue) stable, (⊗ red) unstable against additional vortex creation in the ring component, (◇ red) unstable against line vortex drift out of the trap. Main diagram: $a_{ij}$ set of scattering lengths. Inset: ${\overline{a}}_{ij}$ set of scattering lengths. The grey shaded area indicates the location of the stability region for the $a_{ij}$ values.

Figure 7

(Color online) Small Skyrmion with $N_{\mathrm{tot}}=2\times {10}^6$ and $N_r=0.6$, stabilized against drift with rotation and pinning. Scattering lengths ${\overline{a}}_{ij}$. (a) Color map of line component density (blue, lowest density; red, highest density) in the $x$, $z$ plane. (b) Ring component density. (c) Density profiles of the line (dashed) and ring components (solid) along the $x$-axis ($y=0$, $z=0$). Parameters as for Fig. 1.

Figure 8

(Color online) Skyrmion with $N_{\mathrm{tot}}=9\times {10}^6$ and $N_r=0.5$ for different surface tensions $\sigma$. (solid black) normal $\sigma$: ${\overline{a}}_{ij}$ set of scattering lengths. (dashed red) low $\sigma$: ${\overline{a}}_{11}$ increased by 2%. (dotted-dashed blue) high $\sigma$: ${\overline{a}}_{12}$ increased by 5%. (a) Densities of the line- and ring components across the core ($z=0$, $y=0$). (b) Mach number $(M_1=v_1∕c_{+})$ of the ring component along the core ($x=0$, $y=0$).