Creation and counting of defects in a temperature-quenched Bose-Einstein condensate

We study the spontaneous formation of defects in the order parameter of a trapped ultracold bosonic gas while crossing the critical temperature for Bose-Einstein condensation at different rates. The system has the shape of an elongated ellipsoid, whose transverse width can be varied. For slow enough temperature quenches we find a power-law scaling of the average defect number with the quench rate, as predicted by the Kibble-Zurek mechanism. A breakdown of such a scaling is found for fast quenches, leading to a saturation of the average defect number. We suggest an explanation for this saturation in terms of the mutual interactions among defects.

Figure 1

Relaxation time $\tau ={\tau }_0({\tau }_{\mathrm{Q}}/{\left|t\right|)}^{z\nu }$ (top panels) and correlation length $\xi ={\xi }_0({\tau }_{\mathrm{Q}}/{\left|t\right|)}^{\nu }$ (bottom panels) across the transition at $t=0$ are illustrated for slow (left) and fast (right) quenches. In the KZ theory, the extent of the frozen region is approximated by the freeze-out time $\widehat{t}$, corresponding to the instant when $\tau$ equals the time distance from the transition $|\epsilon /\dot{\epsilon }|$ (dashed lines). The correlation length at such a time provides an estimate for the average defect size $\widehat{\xi }$. The illustration shows that $\widehat{\xi }$ is smaller in case of faster quenches (small ${\tau }_{\mathrm{Q}})$.

Figure 2

Qualitative illustration of the spontaneous formation of vortices or solitons in elongated systems depending on the relative magnitude of the transverse size $\mathrm{\Delta }r$, the domain size $\widehat{\xi }$, and the healing length ${\xi }_l$.

Figure 3

Sequence of experimental absorption pictures of the atomic sample around the transition, occurring at ${\nu }_{\mathrm{c}}\sim 1360\text{kHz}$ for a ramp of 203 kHz/s with $\kappa =10.1$. All these pictures have been taken after a time of flight of 50 ms. At $t=0$ a small condensate fraction of $\sim 1\%$ of the atoms appears in the thermal cloud and grows for $t>0$. In the last picture the condensate appears much more definite, and a defect becomes visible in the form of a vertical stripe in the integrated density distribution. With such a technique it is almost impossible to detect the presence of defects around $t\sim 0$, as they start to become observable only about 100 ms after the transition. Note also that isolated dark pixels are just given by saturation due to electronic noise on the CCD camera and should not be confused with the actual defects of the condensate.

Figure 4

Experimental quench sequence for three ramp speed values. The radio frequency $\nu$ is plotted as a function of time around the transition point $(t=0\text{s})$. A preliminary evaporative cooling stage is the same for all samples (light blue line), then the quench ramp (dark blue line) differs in different experiments. The critical point ${\nu }_{\mathrm{c}}$ of the BEC transition is reported for each quench ramp (cyan squares). The evolution time $t_{\mathrm{e}}$ is kept constant, relative to the transition (shadowed in green). After a time $t_{\mathrm{e}}=250$ ms the sample is released, allowed to expand for a fixed TOF of 120 ms (shadowed in red), and finally observed with absorption imaging (dashed lines). For faster quenches a waiting time is added while keeping a constant RF shield on. The quench times ${\tau }_{\mathrm{Q}}$ in the three cases are 970, 240, and 60 ms.

Figure 5

(a) Measured temperature across $T_{\mathrm{c}}$ for a slow (58 kHz/s), medium (158 kHz/s), and fast (710 kHz/s) quench ramp. A linear fit is shown and the transition region is highlighted. (b) Critical temperature at the BEC transitions as a function of the ramp speed for different ${\omega }_{\mathrm{rad}}$. These values of $T_{\mathrm{c}}$ are used in Eq. (8) for the determination of the quench time.

Figure 6

Average number of defects $\langle N_{\mathrm{d}}\rangle$ as a function of the quench time ${\tau }_{\mathrm{Q}}$ for several transverse confinements and with a fixed evolution time $t_{\mathrm{e}}=250$ ms. Each point with its error bar is calculated by averaging over tens of experimental realizations. A power-law behavior (linear in the log-log scale) is observed for large ${\tau }_{\mathrm{Q}}$, while a saturation effect is present for small ${\tau }_{\mathrm{Q}}$. The lines correspond to the function $\langle N_{\mathrm{d}}\rangle =N_{\mathrm{sat}}{[1+{({\tau }_{\mathrm{Q}}/{\tau }_{\mathrm{Q}}^0)}^{2\alpha }]}^{-1/2}$. The fitting parameters are the saturation number $N_{\mathrm{sat}}$, the KZ power-law exponent $\alpha$, and the characteristic quench time ${\tau }_{\mathrm{Q}}^0$ at which the two regimes interpolate. Their values are given in Table 2.

Figure 7

Power-law exponent $\alpha$ obtained by fitting the data of Fig. 6 for different radial frequencies. The KZ predictions for solitons and vortices in an harmonically trapped 3D condensate, from Table 1, are shown as horizontal lines for comparison.

Figure 8

Average number of defects as a function of the quench time with a fixed ${\omega }_{\mathrm{rad}}=2\pi \times 131$ Hz, for different evolution times. We note that the evolution time does not substantially influence the linear behavior, which can be fitted with a single power-law (dashed purple).

Figure 9

Defect counting statistics. Histograms show the measured occurrence probability of $N_{\mathrm{d}}$ for given intervals of $\langle N_{\mathrm{d}}\rangle$ in the data reported in Fig. 6. The number of experimental runs $N$ considered for each histogram is reported. Histograms are compared to the Poisson distribution which has its mean value equal to the central value of the considered bin. The agreement with the experimental data is good for small values of $\langle N_{\mathrm{d}}\rangle$, i.e., in the power-law regime (upper panels), while it is bad in the saturation regime (lowest two panels).

Figure 10

Predictions for the average defect number versus quench time, including the effects of defect decay during the evolution time after the BEC transition (see text). The dashed line is the KZ power-law scaling as in Fig. 8. The solid curves, from top to bottom, correspond to the average defect numbers expected after an evolution time $t_e=250$, 400, and 750 ms, respectively.