Coherent Light Scattering from a Two-Dimensional Mott Insulator

We experimentally demonstrate coherent light scattering from an atomic Mott insulator in a two-dimensional lattice. The far-field diffraction pattern of small clouds of a few hundred atoms was imaged while simultaneously laser cooling the atoms with the probe beams. We describe the position of the diffraction peaks and the scaling of the peak parameters by a simple analytic model. In contrast to Bragg scattering, scattering from a single plane yields diffraction peaks for any incidence angle. We demonstrate the feasibility of detecting spin correlations via light scattering by artificially creating a one-dimensional antiferromagnetic order as a density wave and observing the appearance of additional diffraction peaks.

Figure 1

Schematics of light scattering. (a) Light diffraction from atoms in a 1D lattice. An incoming wave with wave vector ${\mathbf{k}}_i$ is diffracted under an angle $\theta$. (b) Resulting differential scattering cross section $\frac{d\sigma }{d\theta }(\theta )$, as given by Eq. (2) for our experimental parameters. Shown are the cases for unity filling of the lattice and atom number $N=16$ (solid blue curve) and for a $z$-Néel AFM along one direction ($N=28$), from which only one spin component is detected (dashed green curve). The gray shaded area indicates the opening angle of our imaging system. (c) Experimental setup. Atoms in a 2D optical lattice are illuminated with four in-plane molasses beams. In situ and far-field diffraction images are recorded with a high-resolution optical imaging system.

Figure 2

Light scattering from a 2D MI. (a) Experimental images from the same atomic sample for four different distances $\Delta z$ of the focal plane with respect to the atom position. The lowest image shows the in situ atom number distribution, whereas the upper image shows the far-field diffraction pattern. (b) Simulated diffraction patterns obtained from a 2D numerical model (see text for details), using the reconstructed atom distribution (lower image) from (a). Red arrows indicate the directions of the optical molasses beams.

Figure 3

Analysis of the diffraction patterns. (a)–(c) In situ images ($N=126$, 199, and 279 atoms) and the corresponding far-field diffraction pattern ($\Delta z=40\mu \mathrm{m}$) from the same experimental run, with illumination time 500 ms each. (d) Angular distribution of the differential scattering cross section obtained from cuts in (c) through the diffraction peaks (dark blue, filled symbols) and the background signal (light blue, open symbols). The peak is fitted with a Gaussian (solid red line). (e) Peak height $A$ versus $N$ together with a quadratic fit. (f) Peak width $w$ versus $N$, with a fit to $w\propto 1/\sqrt{N}$. (g) Total background signal ${\sigma }_b$ (light blue, open symbols) as obtained from the constant background in (d) and total signal in the peaks ${\sigma }_p$ (dark blue, filled symbols, scaled by a factor of 10) obtained from the fits to the diffraction peaks (see text).

Figure 4

Light scattering for 1D antiferromagnetic order in the density. (a) The atoms in every second row were removed. (b) Reconstructed atom number distribution from (a). (c) Resulting far-field image ($\Delta z=25\mu \mathrm{m}$) with two diffraction orders in the $x$ direction. (d) Simulated diffraction pattern using the atom number distribution of (b). (e),(f) Angular distribution of the differential scattering cross section obtained from cuts along the $x$ and $y$ directions together with Gaussian fits.