Dynamics of the modified Kibble-Żurek mechanism in antiferromagnetic spin-1 condensates

We investigate the dynamics and outcome of a quantum phase transition from an antiferromagnetic to a phase-separated ground state in a spin-1 Bose-Einstein condensate of ultracold atoms. We explicitly demonstrate double universality in the dynamics within experiments with various quench times. Furthermore, we show that spin domains created in the nonequilibrium transition constitute a set of mutually incoherent quasicondensates. The quasicondensates appear to be positioned in a semiregular fashion, which is a result of the conservation of local magnetization during the postselection dynamics.

Figure 1

Ground-state phase diagram of an antiferromagnetic condensate for magnetization $M=N/2$. We increase $B$ linearly during the time ${\tau }_{\mathrm{Q}}$ to drive the system through a phase transition into a phase-separated state.

Figure 2

Spin domain formation dynamics in a ring-shaped 1D geometry with ring length $L=200\mu$m and ${\omega }_{\perp }=2\pi \times 1000$Hz, for $N=2\times {10}^7$ atoms. The density of the $m_f=0$ component $|{\psi }_0{|}^2$ is shown. The quench time is ${\tau }_{\mathrm{Q}}=28.6$ ms.

Figure 3

Rescaled number of domain seeds $N_{\mathrm{s}}$ and number of domains $N_{\mathrm{d}}$ (Ref. 23) versus rescaled time after $t_{\mathrm{c}}$. Here $t_{\mathrm{c}}$ corresponds to $B=B_{\mathrm{c}}$. The figures demonstrate double universal dynamics during the formation of domain seeds (top), and at long times (bottom), for experiments with different ${\tau }_{\mathrm{Q}}$. We ascribe the deviations of rescaled $N_s$ in the top figure to the technical difficulty of determining the exact number of seeds before they are fully formed. The number of domains is decreased by 2 to account for ground-state phase separation into two domains. The data are averaged over 100 realizations. The parameters are the same as in Fig. 2.

Figure 4

Examples of bubble density profiles for two values of ${\tau }_{\mathrm{Q}}$. While the average distance between the bubbles is very different in the two cases, the size of a single bubble remains approximately the same. The parameters are as in Fig. 2.

Figure 5

Plot of the fraction $x_0$ of the system volume occupied by ${\rho }_0$ domains and the fraction $n_0$ of atoms in the $m=0$ component as a function of time in a single experiment. Here ${\tau }_{\mathrm{Q}}=714$ ms for the main plot, and ${\tau }_{\mathrm{Q}}=28.6$ ms for the inset. Apart from small discrepancies, there is a good agreement with the predicted static value (solid lines). The agreement improves with increasing quench time ${\tau }_{\mathrm{Q}}$. The parameters are as in Fig. 2.

Figure 6

Probability of two neighboring domains being separated by a specified distance in micrometers, in 1000 realizations of the experiments. The solid line is a Gaussian fit. Clearly, there is a regularity in placement of domains. The parameters are as in Fig. 2.

Figure 7

Dynamics of correlation functions $g_0^{\left(1\right)}(x,0,t)$ (top panel) and $g_0^{\left(2\right)}(x,0,t)$ (bottom panel) averaged over ${10}^4$ realizations for ${\tau }_{\mathrm{Q}}=30$ ms, showing the evolution of coherence in the $m_F=0$ spin component. The parameters are as in Fig. 2.

Figure 8

Profiles of correlation functions $g_0^{\left(1\right)}(x,0,t)$ (black line) and $g_0^{\left(2\right)}(x,0,t)$ (red line) at long time (here $t=40$ ms), for ${\tau }_{\mathrm{Q}}=30$ ms averaged over ${10}^4$ realizations. The correlation length can be obtained as the half-width of $g_0^{\left(1\right)}(x,0,t)$. The distance between two maxima of $g_0^{\left(2\right)}(x,0,t)$ corresponds to the average distance between the domains.