Topological defect formation in a quenched ferromagnetic Bose-Einstein condensates

We study the dynamics of the quantum phase transition of a ferromagnetic spin-1 Bose-Einstein condensate (BEC) from the polar phase to the broken-axisymmetry phase by changing magnetic field, and find the spontaneous formation of spinor domain walls followed by the creation of polar-core spin vortices. We also find that the spin textures depend very sensitively on the initial noise distribution, and that an anisotropic and colored initial noise is needed to reproduce the Berkeley experiment [Sadler et al., Nature (London) 443, 312 (2006)]. The dynamics of vortex nucleation and the number of created vortices depend also on the manner in which the magnetic field is changed. We point out an analogy between the formation of spin vortices from domain walls in a spinor BEC and that of vortex-antivortex pairs from dark solitons in a scalar BEC.

Figure 1

Phase diagram of an homogeneous spin-1 BEC with $c_1<0$. In the ferromagnetic phase, the ground state is the $m=1$ state for $p>0$ and the $m=-1$ state for $p<0$, while in the polar phase, the ground state is the $m=0$ state. The ground-state wave function in the broken-axisymmetry phase possesses three nonzero components given in Eq. (8), and the spin vector tilts against the applied magnetic field. The arrow indicates the direction of the quench of the magnetic field.

Figure 2

Magnitude and direction of the spin at $y=0$ for the initial condition in Eq. (26) with $\epsilon =0.05$. The upper panels show transverse spin density $\mid F_{+}\mid$ and the lower panels show its phase $\arg \left(F_{+}\right)$. The gray scale represents the density from 0 to the initial peak value $D_{\mathrm{p}}=n(\mathit{r}=\mathbf{0},t=0)$ for the upper panels, and the phase from $-\pi$ to $\pi$ for the lower panels. The field of view of each panel is $400\times 25\mu \mathrm{m}$.

Figure 3

(Color) (a) Time evolution of $G_T\left(0\right)$ (solid curve) and $G_L\left(0\right)$ (dashed curve) and (b) $G_T(x,z)$ for the initial condition given in Eq. (26) with $\epsilon =0.05$.

Figure 4

Magnitude and direction of the spin for the initial condition given in Eq. (27).

Figure 5

(Color) (a) Time evolution of $G_T\left(0\right)$ (solid curve) and $G_L\left(0\right)$ (dashed curve) and (b) $G_T(x,z)$ for the initial condition given in Eq. (27).

Figure 6

Magnitude and direction of the spin for the initial condition given in Eq. (28) with ${\lambda }_{\text{cutoff}}=60\mu m$.

Figure 7

(Color) (a) Time evolution of $G_T\left(0\right)$ (solid curve) and $G_L\left(0\right)$ (dashed curve) and (b) $G_T(x,z)$ for the initial condition given in Eq. (28) with ${\lambda }_{\text{cutoff}}=60\mu m$.

Figure 8

Time evolution of the system, in which the magnetic field is decreased from $530\text{to}50\mathrm{mG}$ during the first $300\mathrm{ms}$ so that $q$ linearly decreases. The initial condition is the same as in Figs. 6, 7.

Figure 9

Ramp time dependence of the number of spin vortices at $t=200\mathrm{ms}$. The initial state is given in Eq. (28) with ${\lambda }_{\text{cutoff}}=60\mu m$ and the magnetic field is decreased from $530\text{to}50\mathrm{mG}$ so that $q$ is linearly ramped down during $t_{\mathrm{ramp}}$. The plots and error bars represent the average and standard deviation with respect to ten runs of simulations for different random numbers to generate the noise.

Figure 10

Time evolution of the system, in which the magnetic field is suddenly decreased to $400\mathrm{mG}$ at $t=0$. The initial condition is the same as in Figs. 6, 7.

Figure 11

Time evolution of the transverse magnetization $\mid ⟨F_{+}⟩\mid =\mid \int d\mathit{r}{F}_{+}\mid$ for the dynamics in Fig. 2 (solid curve), Fig. 4 (dashed curve), and Fig. 6 (dotted curve).