Relaxation Dynamics in the Merging of $N$ Independent Condensates

Controlled quantum systems such as ultracold atoms can provide powerful platforms to study nonequilibrium dynamics of closed many-body quantum systems, especially since a complete theoretical description is generally challenging. In this Letter, we present a detailed study of the rich out-of-equilibrium dynamics of an adjustable number $N$ of uncorrelated condensates after connecting them in a ring-shaped optical trap. We observe the formation of long-lived supercurrents and confirm the scaling of their winding number with $N$ in agreement with the geodesic rule. Moreover, we provide insight into the microscopic mechanism that underlies the smoothening of the phase profile.

Figure 1

Experimental protocol. (a) Illustration of the experimental sequence. An annular trap is partitioned into $N$ segments of equal length. Uncorrelated BECs are prepared in these segments with random phase differences $\delta {\varphi }_i$, $i=1,\dots ,N$, between adjacent condensates. After merging into a single annular condensate, supercurrents with winding number $\nu \in \mathbb{Z}$ are formed. (b) In situ density distribution in the ring trap for $N=9$ at different times $t$ during the merging. The outer ring has a mean radius of $19.5\mu \mathrm{m}$ and a width of $5\mu \mathrm{m}$. The inner ring serves as a phase reference for the detection as described in the main text. It has a mean radius of $13\mu \mathrm{m}$ and a width of $4\mu \mathrm{m}$. Each image is an average over 5 or 6 experimental realizations. (c) Matter-wave interference after a 2D time of flight (TOF) of 6 ms. The chirality of the pattern and the number of spiral arms reveal the winding number $\nu$ of the supercurrent in the outer ring.

Figure 2

Formation of supercurrents as a function of the number of BECs $N$. (a) Probability distributions $p(\nu )$ for $N=1$, 3 and 9 obtained from $\mathcal{M}=202$, 238, and 388 measurements, respectively. The insets display in situ images before the merging averaged over 4–6 realizations. (b) Measured rms width ${\nu }_{\mathrm{rms}}$ of the probability distributions as a function of $N$. Each data point consists of $\mathcal{M}>200$ independent measurements. The corresponding mean values $\overline{\nu }$ are displayed in the inset. The solid line is the predicted scaling given in Eq. (1). All error bars display the combined uncertainty from the experimental determination of the winding number and the statistical error due to a finite number of measurements $\mathcal{M}$, which was evaluated using a bootstrapping approach.

Figure 3

Relaxation dynamics from $N$ to $N/2$ condensates, when merging them in two successive steps. The in situ images above the main graph illustrate the experimental sequence for $N=12$. Each image is an average over five individual measurements. The main graph depicts our experimental results for $N=12$ (black) and $N=6$ (blue). Each data point consists of $\mathcal{M}>200$ measurements. The corresponding mean values $\overline{\nu }$ are shown in the Supplemental Material [16]. The error bars depict the uncertainty obtained from our finite number of measurements $\mathcal{M}$ and the experimental uncertainty in the determination of the winding numbers. The dashed lines indicate the measured values shown in Fig. 2 and the shaded areas illustrate the corresponding error bars. The solid lines are fits of exponential functions $f_j(t_{\text{wait}})=A_j{\mathrm{e}}^{-t_{\text{wait}}/{\tau }_j}+B_j$, $j=\{6,12\}$, to our data, where ${\tau }_j$ is the only free fit parameter and the other variables are determined by the dashed lines extracted from Fig. 2.

Figure 4

Defect dynamics. (a) In situ density distribution of two line-shaped condensates (first two images) with dimensions $50\mu \mathrm{m}\times 5\mu \mathrm{m}$ before and after the merging (averaged over four individual realizations). The condensates are separated by $3\mu \mathrm{m}$. After merging the condensates in 9.5 ms the system evolves for a variable time $t_{\text{wait}}$. Phase defects are detected by matter-wave interference after TOF (image on the right) [16]. A typical image for $t_{\text{wait}}=0.7\mathrm{ms}$ is depicted on the right. The phase defect at position $y_d$ is highlighted by the dashed line. (b) Position distribution $p(y_d)$ of the phase defects as a function of the waiting time $t_{\text{wait}}$ evaluated from 200 individual measurements. The histograms are normalized by the total number of measurements. Phase dislocations are detected, if the phase difference between neighboring pixels (corresponds to $1.16\mu \mathrm{m}$ in the atomic plane) is larger than $0.3\pi$ [16]. (c) Mean number of phase defects $N_d$ as a function of time. The data were evaluated using a threshold of $0.3\pi$. The shaded area illustrates the sensitivity due to this analysis (upper bound: $0.16\pi$; lower bound: $0.43\pi$).