Defect Saturation in a Rapidly Quenched Bose Gas

We investigate the saturation of defect density in an atomic Bose gas rapidly cooled into a superfluid phase. The number of quantum vortices, which are spontaneously created in the quenched gas, exhibits a Poissonian distribution not only for a slow quench in the Kibble-Zurek (KZ) scaling regime but also for a fast quench, in which case the mean vortex number is saturated. This shows that the saturation is not caused by destructive vortex collisions, but by the early-time coarsening in an emerging condensate, which is further supported by the observation that the condensate growth lags the quenching in the saturation regime. Our results demonstrate that the defect saturation is an effect beyond the KZ mechanism, opening a path for studying critical phase transition dynamics using the defect number distribution.

Figure 1

Spontaneous defect formation in an atomic Bose gas quenched into a superfluid phase. A sample is cooled by linearly decreasing the trap depth for variable quench time $t_q$. Images of the gas for various $t_q$, obtained after a time of flight, where quantum vortices generated during the phase transition are detected by their expanded, density-depleted cores.

Figure 2

Vortex number statistics. (a) Probability distribution of vortex number for various quench times $t_q$. Each distribution is obtained from a collection of 100 measurements for a given $t_q$. The solid lines denote Poissonian curves fit to the data. (b) Mean value ${\overline{N}}_v$ and (c) variance ${\sigma }_v^2$ of the vortex number as functions of $t_q$ on log-log axes. The dashed line in (b) is a model curve fit to the data, where saturation time $t_{\mathrm{sat}}=2.3\mathrm{s}$ (see text for details). The solid lines denote a power-law function fit to the data in the KZ regime of $t_q\ge 1.5t_{\mathrm{sat}}$ (gray circles). Inset in (c) displays the measurement results in plane of ${\overline{N}}_v$ and ${\sigma }_v^2$, and the solid line represents a linear fit to them, yielding ${\sigma }_v^2=1.01(12){\overline{N}}_v$. The error bars in (b) and (c) indicate the standard errors. If the error bars are invisible, they are shorter than the marker size.

Figure 3

(a) VMR of the vortex number as a function of $t_q$. The solid line represents a model curve based on the vortex decay scenario, where the initial mean vortex number is given by the KZ scaling prediction (see text for details). (b),(c) Relaxation of the vortex number distribution. ${\overline{N}}_v$ (solid) and ${\sigma }_v^2$ (open) are displayed as functions of additional holding time $t_r$ on log-linear axes for (b) $t_q=0.9$ and (b) 4 s. Each data point was obtained from 100 realizations of same experiment. The error bars in (a)–(c) are the standard errors of the measured quantities.

Figure 4

Postquench condensate growth. (a) Condensate fraction at the end of the quench (circle, $t_h=0$) and after a hold time of $t_h=1.25\mathrm{s}$ (diamond) as functions of $t_q$ on log-log axes. Inset: condensates at various hold times $t_h$ after a quench with $t_q=1.3\mathrm{s}$. (b),(c) Time evolution of the temperature (open circle) and the condensate fraction (solid circle) of the sample in the quenching for (b) $t_q=0.9$ and (c) 6 s. $t_h/t_q=-1(0)$ corresponds to the start (end) of the quench. The dashed lines are guides to the eyes for the temperature evolution. Each data point in (a)–(c) was obtained from ten realizations of same experiment, except those marked by diamonds in (a), which were from 100 measurements. Error bars indicate standard errors of the mean. If the error bars are invisible, they are shorter than the marker size.