Hydrodynamic generation of skyrmions in a two-component Bose-Einstein condensate

When an obstacle is moved in a superfluid faster than a critical velocity, quantized vortices are generated behind the obstacle. Here we propose a method to create more complicated topological excitations, three-dimensional skyrmions, behind a moving obstacle. We numerically show that, in a two-component Bose-Einstein condensate, component-dependent obstacle potentials can generate skyrmions in the wake, made up of quantized vortex rings in different components that are linked with each other. The lifetime of generated skyrmions can be prolonged by a guiding potential, which enables the formation of a skyrmion train.

Figure 1

(a)–(c) Skyrmion state obtained by imaginary-time propagation of the GP equation starting from Eq. (2), where $g_{12}=g$. (a) Isodensity surface of component 2 ($|{\psi }_2{|}^2=0.5$). (b) Density and phase (inset) profiles of component 1 on the plane of $x=0$. (c) Density and phase (inset) profiles of component 2 on the plane of $z=0$. (d)–(f) Spin state of (a)–(c) rotated by $\pi /2$ using Eq. (7). (d) Isodensity surfaces of both components ($|{\psi }_1{|}^2={\left|{\psi }_1\right|}^2=0.2$). (e) and (f) Density and phase (insets) profiles of components 1 and 2 on the plane of $x=0$. The white arrows indicate the positions of quantized vortex cores. (g) Time evolution of the isodensity surfaces for $g_{12}=0.95g$. The size of the boxes in (a), (d), and (g) is $128\times 128\times 128$ in units of the healing length. See the Supplemental Material for videos of the real-time dynamics in (a), (d), and (g) [50].

Figure 2

(a) Isosurfaces of the obstacle potentials ($V_1=V_2=0.5V_0$) in Eq. (8). The parameters of the potential are $V_0=1, {\delta }_1=-{\delta }_2=3, {\lambda }_1=-{\lambda }_2=\pi /16$, and $R=14$. (b)–(f) Dynamics of skyrmion generation for $g_{12}=0.98g$. The potential in (a) is moved at a velocity $v_z=0.18$, which is linearly ramped down and vanishes at $t=100$. Isodensity surfaces of both components ($|{\psi }_1{|}^2={\left|{\psi }_2\right|}^2=0.2$) are shown, where the densities are smaller inside the surfaces. The frame of reference is moved at $v_z=0.18$ in the $z$ direction. The size of the boxes in (b)–(f) is $128\times 128\times 128$ in units of the healing length. See the Supplemental Material for a video of the dynamics in (b)–(f) [50]. (g) Time evolution of the winding number $W$ defined in Eq. (5). (h) Critical velocity $v_c$ above which skyrmions are generated. The parameters of the potential are $V_0=1, {\delta }_1=-{\delta }_2=3$, and ${\lambda }_1=-{\lambda }_2=\pi /16$.

Figure 3

Sequential generation of skyrmions for $g_{12}=0.95g, v_z=0.16, V_0=1, {\delta }_1=-{\delta }_2=0.5, {\lambda }_1=-{\lambda }_2=\pi /24$, and $R=10$. The strength of the potential $V_0$ is constant. (a)–(e) Uniform system. (f)–(j) Cylindrical potential in Eq. (9) with $R_c=32$ is added. In (a)–(d) and (f)–(i), isodensity surfaces of both components ($|{\psi }_1{|}^2={\left|{\psi }_2\right|}^2=0.2$) are shown. The size of the boxes is $128\times 128\times 256$ in units of the healing length. In (f)–(i), the radius of the cylinder is shown by shading. See the Supplemental Material for videos of the dynamics in (a)–(d) and (f)–(i) [50]. (e) and (j) Time evolution of the winding number $W$ without and with the cylindrical potential.

Figure 4

Growth of a skyrmion from overlapped vortex rings for $g_{12}=0.95g$ at (a) $t=100$, (b) $t=1800$, and (c) $t=2800$. The vortex rings are generated by the potential in Eq. (8) with $V_0=1, {\delta }_1=-{\delta }_2=0.5, {\lambda }_1=-{\lambda }_2=\pi /200, R=14$, and $v_z=0.35$. The potential is linearly ramped down and vanishes at $t=100$. Isodensity surfaces of both components ($|{\psi }_1{|}^2={\left|{\psi }_2\right|}^2=0.2$) are shown. The size of the boxes is $128\times 128\times 128$ in units of the healing length. See the Supplemental Material for a video of the dynamics [50].

Figure 5

(a) Isosurfaces of the obstacle potentials $V_1^{\mathrm{obs}}=V_2^{\mathrm{obs}}=2V_0$ (inner) and $1.05V_0$ (outer) in Eqs. (10, 11, 12) with $R=3a_{\mathrm{ho}}, \lambda =\pi /12, \delta =0.5a_{\mathrm{ho}}, b_1=b_2=0.85$, and $V_0=10\hslash \omega$. (b)–(d) Dynamics of skyrmion generation, where $N_1=N_2=3\times {10}^5$ atoms are confined in an isotropic harmonic potential with a frequency $\omega =2\pi \times 100$ Hz: (b) $\omega t=4$, (c) $\omega t=6$, and (d) $\omega t=18$. The obstacle potentials in (a) are moved at a velocity $v_z=0.5a_{\mathrm{ho}}\omega$ and the strength $V_0$ is linearly ramped down and vanishes at $\omega t=3$. The isodensity surfaces of both components ($|{\psi }_1{|}^2={\left|{\psi }_2\right|}^2={10}^{-4}{N}_j/a_{\mathrm{ho}}^3$) are shown, where the surfaces at $r>6a_{\mathrm{ho}}$ are made transparent. The size of the boxes in (b)–(d) is ${\left(17.92a_{\mathrm{ho}}\right)}^3$. See the Supplemental Material for a video of the dynamics in (b)–(d) [50].