Winding Up of the Wave-Function Phase by an Insulator-to-Superfluid Transition in a Ring of Coupled Bose-Einstein Condensates

We study phase transition from the Mott insulator to superfluid in a periodic optical lattice. Kibble-Zurek mechanism predicts buildup of winding number through random walk of BEC phases, with the step size scaling as a third root of transition rate. We confirm this and demonstrate that this scaling accounts for the net winding number after the transition.

Figure 1

Kinetic hopping energy $K_1=1-C_1\approx \overline{1-\cos \Delta {\theta }_s}\approx \overline{\frac{1}{2}\Delta {\theta }_s^2}$ as a function of rescaled $J/\widehat{J}$ for different ${\tau }_Q$ is seen in (a). When $J\ll 1$, all the plots overlap demonstrating that $\widehat{J}={\tau }_Q^{-2/3}$ is the relevant scale for $J\ll 1$. Individual plots depart from this small-$J$ “common bundle” at $J\simeq 1$, or $J/{\tau }_Q^{-2/3}\simeq {\tau }_Q^{2/3}$, when $K_1=1-C_1$ is expected to stabilize. In (b), we show $K_R\equiv 1-C_R$ at $J=10$ as a function of ${\tau }_Q$ for $R=1,\dots ,5$. Data points for each $R$ were fitted with lines, their slopes giving exponents close to the $\frac{1}{3}$-scaling predicted in Eq. (10) with error bars on their last digits. Size of typical phase step can be estimated as $\overline{\Delta {\theta }_s^2}\simeq K_R/R$ when $K_R\ll 1$, and already this rough estimate yields a good approximation of winding numbers shown in Fig. 2.

Figure 2

Variance of winding number $\overline{W_N^2}$ measured at $J=10$ as a function of ${\tau }_Q$ for lattice sizes $N=512$, 256, 128. Here, point sizes equal error bars. The data points with ${\tau }_Q>2$ and $\overline{W_N^2}>2$ were fitted with the solid lines giving slopes close to the predicted $-\frac{1}{3}$ in Eq. (12). $\overline{W_N^2}$ is shown over a wider range of ${\tau }_Q$ to show the saturation for nearly instantaneous quenches, when ${\tau }_Q<2$, and the crossover to steeper slope when $\overline{W_N^2}<2$. In the inset, we show a rescaled $\overline{W_N^2}/N$ for $N=512$, 256, 128, 32, 8 to demonstrate that $\overline{W_N^2}\sim N$ in the KZM regime of ${\tau }_Q>2$ and $\overline{W_N^2}>2$.

Figure 3

Correlation function $C_R$ for different ${\tau }_Q$ measured at $J=10$ on a lattice of $N=512$ sites. This logarithmic plot demonstrates that $C_R$ is exponential in $R$.