Observation of a 2D Bose Gas: From Thermal to Quasicondensate to Superfluid

We present experimental results on a Bose gas in a quasi-2D geometry near the Berezinskii, Kosterlitz, and Thouless (BKT) transition temperature. By measuring the density profile after time of flight and the coherence length, we identify different states of the gas. We observe that the gas develops a bimodal distribution without long range order. In this regime, the gas presents a longer coherence length than the thermal cloud; it is quasicondensed but is not superfluid. Experimental evidence indicates that we also observe the superfluid transition (BKT transition). For a sufficiently long time of flight, we observe a trimodal distribution when the gas has developed a superfluid component.

Figure 1

Calculated total density (tot), quasicondensate (QC) density, and superfluid (SF) density at the center of the cloud in a 2D trap versus the total number of atoms $N$ normalized to $N_{\mathrm{crit}}$, the critical number for BEC of a 2D trapped, noninteracting gas. Insets: representative images and corresponding density profiles, measured along a line passing through the center of the cloud, for different $N$ after 5 ms TOF. The dot-dashed (red) line is a fit to the sum of two Gaussians, while the solid (blue) line is the wider Gaussian.

Figure 2

Width of the narrow part of the cloud from fits of 5 ms TOF images by two Gaussians (see insets of Fig. 1) versus the fitted peak 2D density, for two different temperatures. The intersection of linear fits to two portions of the data is used to determine a transition point. Inset: 2D “in situ” peak density [25] at the transition point as a function of the temperature. The dashed line is the theoretical value $n_{\mathrm{BKT}}{\lambda }^2=\ln (380/\tilde{g})$ and the solid line is the same value corrected using a 3D model [13]. We add a $\pm 15\%$ uncertainty [27] primarily due to the disagreement of the different methods used for calibration of the temperature and density.

Figure 3

Cross section of the density profile of the cloud after 10 ms of TOF, fitted with the sum of three Gaussians for $T\approx 100\mathrm{nK}$. Left: for $n<n_{\mathrm{BKT}}$ ($T>T_{\mathrm{BKT}}$); right: for $n>n_{\mathrm{BKT}}$ ($T<T_{\mathrm{BKT}}$).

Figure 4

Left: Schematic of the atom interferometer used to measure the spatial coherence of the 2D atomic cloud. The Raman laser beams have frequencies ${\nu }_i$ and corresponding wave vectors ${\mathbf{k}}_i$. Right: Images of atoms in the $F=2$ state after the interferometer sequence for two different delays between Raman pulses (upper: $\tau =7.5\mu \mathrm{s}$, lower: $\tau =17.5\mu \mathrm{s}$). For the longer delay, the interference of the thermal cloud is washed out and there are only fringes from the superfluid region, which has a longer coherence length.

Figure 5

Contrast of the interference fringes as a function of the separation of the two clouds in the interferometer. Each curve corresponds to an average over many data sets where the maximum density of atoms falls within a chosen range. Inset: Corresponding density profiles of the atomic cloud. Lines (solid, dotted, dashed, dot-dashed) correspond, respectively, to density ranges of 2–4, 5–8, 10–12, $13–19\mathrm{atoms}/\mu {\mathrm{m}}^2$.