Universal Early Coarsening of Quenched Bose Gases

We investigate the early coarsening dynamics of an atomic Bose gas quenched into a superfluid phase. Using a two-step quench protocol, we independently control the two cooling rates during and after passing through the critical region, respectively, and measure the number of quantum vortices spontaneously created in the system. The latter cooling rate regulates the temperature during the condensate growth, consequently controlling the early coarsening dynamics in the defect formation. We find that the defect number shows a scaling behavior with the latter cooling rate regardless of the initial cooling rate, indicating universal coarsening dynamics in the early stage of condensate growth. Our results demonstrate that early coarsening not only reduces the defect density, but also affects its scaling with the quench rate, which is beyond the Kibble-Zurek mechanism.

Figure 1

Early coarsening in a quenched Bose gas. As a thermal Bose gas is cooled into a superfluid phase, it experiences an early coarsening period (green), after passing through the critical region (blue) but before the order parameter has grown significantly ($t_{\mathrm{fr}}<t<t_d$), during which the spatial fluctuations of the system are coarsened, reducing the defect creation probability. $T_c$ denotes the critical temperature. In the critical region, the system’s dynamics is effectively frozen due to the divergence of its relaxation time at the critical point. In the upper row, the system’s quench evolution is illustrated, where the boundaries of spatial domains of the symmetry-broken phase are indicated by dashed lines and quantum vortices are denoted by the white circles.

Figure 2

Two-step quench experiment. (a) Schematic of the quench protocol. The trap depth $U$ is linearly lowered from $U_i$ to $U_m$ for time ${\tau }_1$ with rate $r_1$ and from $U_m$ to $U_f$ for ${\tau }_2$ with rate $r_2$. $U_c$ denotes the critical trap depth for Bose-Einstein condensation. After a hold time ${\tau }_h$, a time-of-flight image is taken to detect created vortices. (b) For a trapped sample, the local critical temperature $T_{c,\text{local}}$ spatially varies over the sample. The inner (outer) region $A$ ($B$) undergoes a phase transition for $U>U_m(<U_m)$. The inset shows images of samples in equilibrium at $U=U_m$ (left) and $U=U_f$ (right). (c) Mean vortex number ${\overline{N}}_v$ as a function of the quench rate on log-log axes. Open circles denote the data obtained with $r_1=r_2$, and red (blue) solid circles show the data for fixed $r_2(r_1)=0.08{\mathrm{s}}^{-1}$. The solid line shows a power-law function with exponent ${\alpha }_{\mathrm{KZ}}=2.6$, fit to the data with $r_1=r_2$ in the scaling regime. Each data point in the blue and red (black) curves was obtained from 30 (20) realizations of the same experiment. The error bars indicate the standard errors of the mean. If the error bars are invisible, they are smaller than the marker size. (d) Representative images of samples for various $r_1$ and $r_2$, displaying quantum vortices by their expanded, density-depleted cores.

Figure 3

Mean vortex number ${\overline{N}}_v$ as a function of $r_1$ and $r_2$ on three-dimensional log axes. Each data point was obtained from 20 realizations of the same experiment, except one with $r_{1(2)}=r_m$ which was obtained from 30 measurements [the same data in Fig. 2]. The error bars indicate the standard errors of the mean.

Figure 4

Universal scaling of early coarsening. For a given value of $r_2$, $\mathcal{D}(r_1,r_2)$ denotes the increase in ${\overline{N}}_v$ as $r_1$ increases from $r_m$ [Eq. (2)]. (a) $\mathcal{D}$ as functions of $r_2$ for four different $r_1$ on log-log axes and (b) the same curves after scaling (see the text for details). The inset in (b) presents a universal curve $f({\tilde{r}}_2)$ (${\tilde{r}}_2=r_2/r_m$) obtained from averaging the data in (b), and the solid line shows a power-law function with exponent $\beta =1.3$, fit to the data in the scaling regime (${\tilde{r}}_2<2.5$). (c) $\mathcal{D}$ as functions of $r_1$ for four different $r_2$ on log-log axes and (d) the same curves after divided by $f({\tilde{r}}_2)$. The inset in (d) presents $N_A(r_1,r_m)$ calculated from the data in (d) (see the text for details), and the dashed lines are power-law guide lines. The error bars in (a)–(d) are the standard errors of the measured quantities.