Mott-Insulator Transition in a Two-Dimensional Atomic Bose Gas

Cold atoms in periodic potentials are versatile quantum systems for implementing simple models prevalent in condensed matter theory. Here we realize the 2D Bose-Hubbard model by loading a Bose-Einstein condensate into an optical lattice, and study the resulting Mott insulator. The measured momentum distributions agree quantitatively with theory (no adjustable parameters). In these systems, the Mott insulator forms in a spatially discrete shell structure which we probe by focusing on correlations in atom shot noise. These correlations show a marked dependence on the lattice depth, consistent with the changing size of the insulating shell expected from simple arguments.

Figure 1

Results at three values of $t/U$: $40\times {10}^{-3}$, $4\times {10}^{-3}$, and $0.5\times {10}^{-3}$. (a) Atom density versus momentum. (b) Noise correlations versus momentum difference. Each displayed image was averaged from $\approx 60$ raw images; to reduce technical noise, the displayed correlation data was averaged with itself, rotated by 90°.

Figure 2

Solid symbols: measured first order coefficient $\alpha$ versus $t/U$. The overlayed red (or gray) dashed line shows the prediction, $\alpha =8t/U$. (These data and the fit have been displaced for clarity: vertically as indicated, and by about 2% horizontally.) Empty symbols: averaged second order coefficient $\beta$ versus $t/U$. The associated red (or gray) dashed line is the prediction, $\beta =90(t/U{)}^2$. The vertical dotted line indicates the expected location of the 2D SF-MI transition [17, 18].

Figure 3

Cross sections of normalized quasimomentum distributions (along $\widehat{x}+\widehat{y}$, and offset for clarity) at three values of $t/U$. The data are plotted along with the theoretical profile (solid lines) [28]. The dashed lines (not visible in the bottom trace) reflect the uncertainty in the theory resulting from the single-shot $\pm 0.5E_R$ uncertainty in the lattice depth.

Figure 4

Average area of the noise-correlation peaks expressed in units of $k_R^2/N_T$. The dashed line, denoting $A$ as calculated in our LDA model, was scaled by 0.45 to lie upon the data (see text). Inset: the solid symbols indicate the measured peak width, $\delta$, in units of $k_R$. The dashed line is the expected width from our LDA computation. The solid line shows the modeled $\delta$ including the imaging resolution [27] of $0.05k_R$. In both cases, the vertical dotted line shows the expected location of the 2D SF-MI transition [17, 18], and the uncertainties reflect the statistical uncertainty of the fit due to background noise.