Condensate Fraction in a 2D Bose Gas Measured across the Mott-Insulator Transition

We realize a single-band 2D Bose-Hubbard system with Rb atoms in an optical lattice and measure the condensate fraction as a function of lattice depth, crossing from the superfluid to the Mott-insulating phase. We quantitatively identify the location of the superfluid to normal transition by observing when the condensed fraction vanishes. Our measurement agrees with recent quantum Monte Carlo calculations for a finite-sized 2D system to within experimental uncertainty.

Figure 1

Momentum distributions and cross sections at $U/t=4(1)$, 8(1), and 20(2). Each row shows a single momentum distribution normalized by the total atom number; the lines in the top right panel indicate trajectories along which four cross sections were taken. The left panel shows the average of these four sections (black solid line); the red dashed lines denote the fit to the bimodal distribution.

Figure 2

Condensed fraction $f$ and $\sigma$ vs $V$ (bottom axis) or $U/t$ (top axis). The dots denote values determined from 2D fits to the full momentum distribution: small dots result from one image and the large dots indicate data averaged over about 20 separate images. The uncertainties are their root mean squared variation, and are indicative of the single-image uncertainties. (a) Condensate fraction. The red dashed line is computed from our MFT model. (b) Fit parameter $\sigma$. At low $U/t$ $\sigma$ is nearly constant (blue dashed line), from which we infer an initial temperature $k_{B}T\approx 2t$. At large $U/t$ $\sigma$ monotonically increases, consistent with predictions of perturbation theory in the MI phase (red dashed line).

Figure 3

Quasimomentum width vs $V$ (bottom axis) or $U/t$ (top axis). The symbols denote the average FWHM of the quasimomentum distribution along the axes of highest symmetry (top: averaged along $\widehat{x}+\widehat{y}$ and $\widehat{x}\mathrm{\text{−}}\widehat{y}$; bottom: averaged along $\widehat{x}$ and $\widehat{y}$). The small and large dots and uncertainties are as explained at Fig. 2. The red dashed line is the horizontally displaced RPA momentum width, as discussed in the text, and the vertical gray line denotes the location of the SF-normal transition identified from the sudden increase in $\Gamma$.