Critical Point of an Interacting Two-Dimensional Atomic Bose Gas

We have measured the critical atom number in an array of harmonically trapped two-dimensional (2D) Bose gases of rubidium atoms at different temperatures. We found this number to be about 5 times higher than predicted by the semiclassical theory of Bose-Einstein condensation (BEC) in the ideal gas. This demonstrates that the conventional BEC picture is inapplicable in an interacting 2D atomic gas, in sharp contrast to the three-dimensional case. A simple heuristic model based on the Berezinskii-Kosterlitz-Thouless theory of 2D superfluidity and the local density approximation accounts well for our experimental results.

Figure 1

Experimental setup. Left: A one-dimensional optical lattice is used to split a magnetically trapped 3D BEC (transparent ellipsoid) into a small array of 2D clouds. Right: Contour lines of the total (magnetic and light) potential $V(x,z)$ in the $y=0$ plane for the lattice phase such that the two central planes are symmetric with respect to the trap center.

Figure 2

Phase transition in a rubidium 2D gas. 2D clouds confined parallel to the $xy$ plane are released from an optical lattice and the density distribution is recorded by absorption imaging along $y$ after $t=22\mathrm{ms}$ of time of flight. The measured line densities $\overline{\rho }(x)$ (●) for an atom number just below (left) and just above (right) the critical number are displayed together with bimodal fits (solid lines). The dashed line in the left panel shows the expected distribution of the 2D ideal gas at the threshold of conventional BEC in our potential at the same temperature ($T=92\mathrm{nK}$). The dotted line in the right panel indicates the Gaussian part of the bimodal distribution.

Figure 3

Measurement of the critical point. The number of atoms in the Thomas-Fermi part of the bimodal distribution $N_0$ (⋄) is plotted as a function of the total atom number $N$. The solid line shows the linear fit we use to determine $N_c$, and the dashed line is its extrapolation. For comparison, the interference amplitude $|\widehat{\rho }(0,k_0)|$ (●) is also displayed as a function of $N$. It shows the same threshold $N_c$ within our experimental precision. The inset shows that the temperature deduced from the Gaussian part of the fit is to a good approximation constant for all data points. Horizontal and vertical dashed lines indicate the average temperature and the critical atom number, respectively. The solid line marks the region used to determine the average temperature $T=92(6)\mathrm{nK}$ close to the transition. Each data point is based on 5–10 images, all error bars represent standard deviations.

Figure 4

Critical point in an interacting 2D gas. The critical atom number $N_c$ (●) is measured at four different temperatures $T$. Displayed error bars are statistical. The solid line shows the ideal 2D gas BEC prediction $N_{c,\mathrm{id}}^{\mathrm{multi}}$. The dashed line is the best empirical fit to the data, which gives $N_c=\alpha N_{c,\mathrm{id}}^{\mathrm{multi}}$ with $\alpha =5.3(5)$.

Figure 5

Coherence and density profile below the transition. Interference of 2D clouds is used to compare the coherent part of the cloud with the part following the Thomas-Fermi density distribution. Left: Interference patterns in the $xz$ plane (see example in inset) are Fourier transformed along the expansion axis $z$ and averaged over ten images taken under identical conditions, to obtain $|\widehat{\rho }(x,k_z)|$. Right: Within experimental precision, fits to the total density profile $|\widehat{\rho }(x,0)|$ and the interference amplitude profile $|\widehat{\rho }(x,k_0)|$ give the same Thomas-Fermi diameter $2R_{\mathrm{TF}}$, indicated by the dashed lines. The weak second harmonic peak at $k_z=2k_0$ reveals small occupation of the outer lattice planes.