Phases of a Two-Dimensional Bose Gas in an Optical Lattice

Ultracold atoms in optical lattices realize simple condensed matter models. We create an ensemble of $\approx 60$ harmonically trapped 2D Bose-Hubbard systems from a ${}^{87}\mathrm{Rb}$ Bose-Einstein condensate in an optical lattice and use a magnetic resonance imaging approach to select a few 2D systems for study, thereby eliminating ensemble averaging. Our identification of the transition from superfluid to Mott insulator, as a function of both atom density and lattice depth, is in excellent agreement with a universal state diagram [M. Rigol et al., Phys. Rev. A 79 053605 (2009)] suitable for our trapped system. In agreement with theory, our data suggest a failure of the local density approximation in the transition region.

Figure 1

State diagram for a harmonically trapped 2D Bose gas. The blue line shows the QMC predicted [7] first appearance of MI. The transition was measured at various $N_{2\mathrm{D}}$ from $f(U/t)$ data (e.g., of Fig. 4, whose data were taken along the green dashed path). The ovals denote the measured transition boundary; their sizes represent the uncertainties in $\tilde{\rho }$ and $(U/t{)}_c$. The small circles indicate individual measurements and are shaded according to the side of the transition on which they are. The yellow dashed line is a fit to the measured boundary for $\tilde{\rho }>20$; this nonvertical [${\theta }_{\exp }=85.5(27)°$ from horizontal], suggests a breakdown of the LDA.

Figure 2

(a) Our 3D BEC is divided into 2D systems by an optical lattice in the presence of a linear magnetic field gradient, both aligned along the $\widehat{z}$ direction. We use a MRI technique to selectively address a small subset of 2D systems. (b) Matter-wave focusing is used to better resolve the SF phase of the 2D Bose gas; shown is a schematic of the focusing of a single 2D system after free evolution during TOF. (c)–(d) We compare the measured atom distribution (approximating the momentum distribution) of the ensemble of $\approx 60$ 2D systems without focusing (c) with that of the addressed 2D systems in the $m_{\mathrm{F}}=0$ sublevel with focusing (d), both with an $\widehat{x}-\widehat{y}$ lattice at $9.5E_R$.

Figure 3

Density profile $n(z)$ for: (a) a 3D BEC and (b) an ensemble of 2D systems. The atom number calculated from the in situ T-F radius $R_z=8.2(2)\mu \mathrm{m}$ is $N_{\mathrm{T}\mathrm{\text{−}}\mathrm{F}}=1.8(4)\times {10}^5$. The vertical dashed lines indicate the T-F radius from our fit. Continuous lines show a fit to the in situ 1D density profile $n(z)$. The temperature of the selected 2D systems [squares in (b)] is displayed on the right axis, as a function of position along $\widehat{z}$. On average $T=15(3)\mathrm{nK}$.

Figure 4

2D condensate fraction for $N_{2\mathrm{D}}\approx 3500$ measured through the SF to MI transition. The red dashed curves are described in the text. The insets (a)–(c) display the average momentum distribution $n(k)=[n_x(k)+n_y(k)]/2$ at different $U/t$, where $n_x(k)$ is the momentum distribution integrated over $y$ and likewise for $n_y(k)$. We identify the formation of the first MI region at $U/t=21(2)$. $f\approx 0$.