Crystallized half-skyrmions and inverted half-skyrmions in the condensation of spin-1 Bose gases with spin-orbit coupling

The nonequilibrium dynamics of a rapidly quenched spin-1 Bose gas with spin-orbit coupling is studied. By solving the stochastic projected Gross-Pitaevskii equation, we show that crystallization of half-skyrmions (merons), can occur in a spinor condensate of ${}^{87}$Rb. The stability of such a crystal structure is analyzed. Likewise, inverted half-skyrmions can be created in a spin-polarized spinor condensate of ${}^{23}$Na. Our studies provide a chance to explore the fundamental properties of skyrmionlike matter.

Figure 1

Snapshots of the density profiles, (a) $|{\Psi }_{-1}{|}^2$, (b) $|{\Psi }_0{|}^2,$ (c) $|{\Psi }_1{|}^2$, and (d) ${\left|\Psi \right|}^2$, of a spinor BEC of ${}^{87}$Rb during quench. The strengths of the SOC are $V_x=1$, $V_y=2$ and the final temperature is $k_{B}T=2.$ The axes of directions are indicated in the bottom right corner. The rightmost column shows the phase profile of equilibrium state. The final particle numbers are $N_{\pm 1}\approx 7.44\times {10}^3,$ $N_0\approx 1.45\times {10}^4$.

Figure 2

Snapshots of the $\widehat{\mathbf{s}}$ texture caused by the results shown in Fig. 1. Arrays of spin vortices (merons) of same handedness appear as domain walls in (b), which drift away towards the periphery of the condensate as shown in (c) and (d).

Figure 3

The equilibrium density profiles, (a) $|{\Psi }_{-1}{|}^2,$ (b) $|{\Psi }_0{|}^2,$ (c) $|{\Psi }_1{|}^2$, and (d) the $\widehat{\mathbf{s}}$ texture, of the spinor BEC of ${}^{87}$Rb at $k_{B}T=10$. The strengths of the SOC are $V_x=1$, $V_y=2$. The final particle numbers are $N_{\pm 1}\approx 7.95\times {10}^3,$ $N_0\approx 1.46\times {10}^4$.

Figure 4

Snapshots of the density profiles, (a) $|{\Psi }_{-1}{|}^2,$ (b) $|{\Psi }_0{|}^2,$ (c) $|{\Psi }_1{|}^2$, and (d) ${\left|\Psi \right|}^2$, of a spinor BEC of ${}^{87}$Rb during quench. The strengths of the SOC are $V_x=V_y=V=1.8$ and the final temperature is $k_{B}T=2$. The axes of directions are indicated in the bottom right corner. The rightmost column shows the phase profiles of equilibrium state. The final particle numbers are $N_{\pm 1}\approx 7.21\times {10}^3$, $N_0\approx 1.42\times {10}^4$.

Figure 5

(a) The three-dimensional orientation for the equilibrium $\widehat{\mathbf{s}}$ texture associated with the results of Fig. 4. The color of the arrow indicates the magnitude of $s_z$. (b) The topological charge density $\sigma$ of the equilibrium spin texture. (c) The associated spin-density distribution $\left|\mathbf{S}\right(\mathbf{r}\left)\right|.$

Figure 6

The equilibrium density profiles, (a) $|{\Psi }_{-1}{|}^2,$ (b) $|{\Psi }_0{|}^2,$ and (c) $|{\Psi }_1{|}^2$, of a spinor BEC of ${}^{87}$Rb at $k_{B}T=10$ with $V_x=V_y=V=1.8$. The densities of all components fluctuate considerably in comparison with those in the lower-temperature case shown in Fig. 4. The final particle numbers are $N_{\pm 1}\approx 7.66\times {10}^3$, $N_0\approx 1.41\times {10}^4$.

Figure 7

(a) The three-dimensional orientation for the equilibrium $\widehat{\mathbf{s}}$ texture associated with the results of Fig. 6. The color of the arrow indicate the magnitude of $s_z$. (b) The topological charge density $\sigma$ of the equilibrium spin texture. (c) The associated spin-density distribution $\left|\mathbf{S}\right(\mathbf{r}\left)\right|$.

Figure 8

The corresponding equilibrium density profiles in momentum space for the SVL states shown in (a) Fig. 4 and (b) Fig. 6, respectively.

Figure 9

Snapshots of the equilibrium density profiles, (a) $|{\Psi }_{-1}{|}^2,$ (b) $|{\Psi }_0{|}^2,$ (c) $|{\Psi }_1{|}^2$, and (d) ${\left|\Psi \right|}^2$, for a spin-polarized spinor BEC of ${}^{23}$ Na. The axes of directions are indicated in the bottom right corner. The strengths of the SOC are $V_x=V_y=V=1.5$ and the final temperature is $k_{B}T=2$. The rightmost column shows the phase profiles of equilibrium state. The final particle numbers are $N_1\approx 9.26\times {10}^4$, $N_0\approx 1.09\times {10}^4,N_{-1}\approx 1.01\times {10}^3$, corresponding to a magnetization ratio of $M/N\approx 0.87$.

Figure 10

The equilibrium $\widehat{\mathbf{s}}$ texture and the associated spin-density distribution $\left|S\right(\mathbf{r}\left)\right|$, for different numbers of inverted merons in a spinor BEC of ${}^{23}$Na at $k_{B}T=2.$ The results for a single inverted meron are shown in (a) and (b), while those for the three inverted merons are shown in (c) and (d). The final particle numbers are $N_1\approx 8.32\times {10}^4,$ $N_0\approx 1.95\times {10}^4,$ $N_1\approx 2.21\times {10}^3,$ corresponding to a magnetization ratio of $M/N\approx 0.77$.

Figure 11

(a) Schematic plot for the stereographic projection of the spin textures onto the two-dimensional sphere $S^2$ for (a) meron and (b) inverted meron.

Figure 12

The equilibrium density profiles, (a) $|{\Psi }_{-1}{|}^2$, (b) $|{\Psi }_0{|}^2$, (c) $|{\Psi }_1{|}^2$, and (d) the $\widehat{\mathbf{s}}$ texture, of a spinor BEC of ${}^{23}$Na with $V_x=1$, $V_y=2$, and $k_{B}T=10.$ The final particle numbers are $N_1\approx 9.06\times {10}^4,$$N_0\approx 1.3\times {10}^4,N_1\approx 2.24\times {10}^3$, corresponding to a magnetization ratio $M/N\approx 0.84$.