Kibble-Zurek mechanism in a quenched ferromagnetic Bose-Einstein condensate

It is shown that spin vortices are created through the Kibble-Zurek mechanism in the quantum phase transition of a spin-1 ferromagnetic Bose-Einstein condensate when the applied magnetic field is quenched to below a critical value. It is also shown that the spin correlation functions have finite correlation lengths, and that the magnetization at widely separated positions grows in random directions, resulting in spontaneous creation of spin vortices. We numerically confirm the scaling laws that the winding number of spin vortices is proportional to the square root of the length of a closed path and, for a slow quench, is proportional to ${\tau }_{\text{Q}}^{-1∕6}$ with ${\tau }_{\text{Q}}$ being the quench time. The relevance of spin conservation to the Kibble-Zurek mechanism is discussed.

Figure 1

(Color online) (a) Time evolution of the autocorrelation function given by Eq. (50) for an instantaneous quench to $q=0$ and $q=q_{\text{c}}∕2$ with 1D ring geometry. (b) Magnitude of the normalized magnetization $\left|F_{+}\right|∕\rho$ (solid curve, left axis) and direction of the magnetization $\text{arg}{F}_{+}$ (dashed curve, right axis) at $t=70\text{ms}$ for $q=0$ and (c) for $q=q_{\text{c}}∕2$. The radius of the ring is $R=50\mu \text{m}$, the atomic density is $n=2.8\times {10}^{14}{\text{cm}}^{-3}$, and the number of atoms is $N={10}^6$.

Figure 2

(Color online) (a) Numerically obtained correlation function defined by Eq. (52) at $t=70\text{ms}$ after $q$ is instantaneously quenched to $q=0$ and $q=q_{\text{c}}∕2$ (solid curves). The dashed curves are theoretical fits using Eqs. (24, 26). The parameters are the same as those in Fig. 1. (b) Dependence of the variance of the spin winding number on the radius of the ring $R$ at $t=70\text{ms}$. The number of atoms is related to $R$ as $N={10}^6\times R\left[\mu \text{m}\right]∕50$. The dashed lines are proportional to $R$. The inset shows the time dependence of ${⟨w^2⟩}_{\text{avg}}$ for $R=50\mu \text{m}$, where only ${⟨w^2⟩}_{\text{avg}}<30$ is shown since the random initial phases make the winding numbers huge. The data in (a) and (b) are averages over 1000 runs of simulations for different initial states produced by random numbers. The error bars in (b) represent the 95% confidence interval of the ${\chi }^2$ distribution.

Figure 3

(Color online) Dependence of the variance of the spin winding number $w$ on quadratic Zeeman energy $q$. The winding number is calculated as $t=70\text{ms}$ after an instantaneous quench. The parameters are the same as those in Fig. 1 except for $q$. The dashed line is proportional to ${\left(1-2q∕q_{\text{c}}\right)}^{3∕2}$. The plots show the averages over 1000 runs of simulations for different initial states produced by random numbers. The error bars represent the 95% confidence interval of the ${\chi }^2$ distribution.

Figure 4

(Color) Dependence of the variance of the spin winding number on quench time ${\tau }_{\text{Q}}$, where $q$ is quenched according to Eq. (30). The time at which each plot is taken depends on ${\tau }_{\text{Q}}$ as $t=4{\tau }_{\text{Q}}^{1∕3}{\left(\hslash ∕q_{\text{c}}\right)}^{2∕3}$, as shown by the arrows in the inset. The radius of the ring is $R=400\mu \text{m}$, the atomic density is $n=2.8\times {10}^{14}{\text{cm}}^{-3}$, and the number of atoms is $N=8\times {10}^6$. The dashed line is proportional to ${\tau }_{\text{Q}}^{-1∕3}$. The inset shows the time evolution of ${⟨w^2⟩}_{\text{avg}}$, where only ${⟨w^2⟩}_{\text{avg}}<20$ is shown since the random initial phases make the winding numbers huge. The data are averages over 1000 runs of simulations for the different initial states produced by random numbers. The error bars represent the 95% confidence interval of the ${\chi }^2$ distribution.

Figure 5

(Color) (a) Time evolution of the autocorrelation function given by Eq. (56) for an instantaneous quench of $q$ with 2D disk geometry. The radius of the disk is $R_{\text{w}}=100\mu \text{m}$, the atomic density is $n=2.8\times {10}^{14}{\text{cm}}^{-3}$, and the number of atoms is $N={10}^7$. (b) Profiles of the magnetization $\left|F_{+}\right|$ (upper) and its direction $\text{arg}{F}_{+}$ (lower) for $q=0$ and (c) for $q=q_{\text{c}}∕2$. The size of each panel is $200\times 200\mu {\text{m}}^{\text{2}}$.

Figure 6

(Color) (a) Spin correlation function defined by Eq. (58) at $t=100\text{ms}$ after $q$ has been instantaneously quenched to $q=0$ and (b) to $q=q_{\text{c}}∕2$. (c) Variance of the winding number along the circumference of a circle of radius $R$, where the data are taken at $t=100\text{ms}$. The blue and green lines are, respectively, proportional to $R$ and $R^2$. In (a)–(c) the parameters are the same as those in Fig. 5, and the data represent averages over 1000 runs of simulations for the different initial states produced by random numbers. The error bars in (c) represent the 95% confidence interval of the ${\chi }^2$ distribution.

Figure 7

(Color) (a) Variance of the spin winding number versus quench time ${\tau }_{\text{Q}}$ for a 2D disk geometry, where $q$ is quenched according to Eq. (30). The inset shows the time evolution of ${⟨w^2⟩}_{\text{avg}}$, where only ${⟨w^2⟩}_{\text{avg}}<10$ is shown since the random initial phases make the winding numbers huge. The time at which each plot is taken depends on ${\tau }_{\text{Q}}$ as $t=4{\tau }_{\text{Q}}^{1∕3}{\left(\hslash ∕q_{\text{c}}\right)}^{2∕3}$, as shown by the arrows in the inset. The dashed line is proportional to ${\tau }_{\text{Q}}^{-1∕3}$. The radius of the disk is $R_{\text{w}}=400\mu \text{m}$ and the closed path for taking the winding number is $R=320\mu \text{m}$. The atomic density is $n=2.8\times {10}^{14}{\text{cm}}^{-3}$ and the number of atoms is $N=1.6\times {10}^8$. The data are averages over 1000 runs of simulations for the different initial states produced by random numbers. The error bars represent the 95% confidence interval of the ${\chi }^2$ distribution.