Figure 1

Experimental system. (a) 2D optical lattice intensity profile. A lattice of Josephson-coupled BECs is created in the white-shaded area. The central box marks the basic building block of our system, the double-well potential shown in (b). The barrier height $V_{\mathrm{OL}}$ and the number of condensed atoms per well $N_{\mathrm{well}}$ control the Josephson coupling $J$, which acts to lock the relative phase $\Delta \varphi$. A cloud of uncondensed atoms at temperature $T$ induces thermal fluctuations and phase defects in the array when $J<T$. (c) Experimental sequence: A BEC (i) is loaded into the optical lattice over 10 s, suppressing $J$ to values around $T$. We allow 2 s for thermalization. To probe the system, we ramp off the lattice on a faster time scale $t_r$ [23] and take images of the recombined condensate. When $J$ is reduced below $T$ (ii)–(vii), vortices (dark spots) appear as remnants of the thermal fluctuations in the array.Reuse & Permissions

Figure 2

Quantitative study of vortices. The areal density of vortices is quantified by the plotted $\mathcal{D}$ defined in the text. $\mathcal{D}$ is extracted only from the central 11% of the condensate region [circle in inset (a)] to minimize effects of spatial inhomogeneity. (b) $\mathcal{D}$ vs $J$ for two data sets with distinct cold and hot temperatures. Each point represents one experimental cycle. The increase in $\mathcal{D}$ with decreasing $J\lesssim 100\mathrm{nK}$ signals the spontaneous appearance of vortices, while the “background” $\mathcal{D}\lesssim 0.01$ for $J\gtrsim 200\mathrm{nK}$ is not associated with vortices. Vortices clearly proliferate at larger $J$ for the hot data, indicating thermal activation as the underlying mechanism. The large scatter in $\mathcal{D}$ at low $J$ is due to shot noise on the small average number of vortices in the central condensate region. (c) Same data as in (b), but averaged within bins of size $\Delta [\log (J)]=0.15$. Error bars of $\mathcal{D}$ are standard errors. (d) Same data as (b), but plotted versus $J/T$. Cold and hot data sets almost overlap on what appears to be a universal vortex activation curve, as confirmed by averaging [(e)], revealing the underlying competition of $J$ and $T$.Reuse & Permissions

Figure 3

(a) Vortex density $\mathcal{D}$ probed at different optical lattice ramp-down time scales $\tau$. A slow ramp provides time for tightly bound vortex-antivortex pairs to annihilate, allowing selective counting of loosely bound or free vortices only, whereas a fast ramp probes both free and tightly bound vortices. A fit to the vortex activation curve determines its midpoint $(J/T{)}_{50\%}$, its 27%–73% width $\Delta (J/T{)}_{27–73}$, and the limiting values ${\mathcal{D}}_{<}$ (${\mathcal{D}}_{>}$) well below (above) $(J/T{)}_{50\%}$. (b) A down-shift in $(J/T{)}_{50\%}$ is seen for slow ramps, consistent with the occurrence of loosely bound or free vortices at lower $J/T$ only. (c) Mapping between ramp-down time scale $\tau$ and estimated size of the smallest pairs surviving the ramp (upper axis). The difference ${\mathcal{D}}_{<}-{\mathcal{D}}_{>}$ measures the number of vortices surviving the ramp (right axis). Comparison to simulated vortex distributions yields a size estimate of the smallest surviving pairs (upper axis). Inset: Smallest possible pair sizes in a hexagonal array: I: $d/\sqrt{3}$; II: $d$; III: $2d/\sqrt{3}$.Reuse & Permissions