Bragg Scattering as a Probe of Atomic Wave Functions and Quantum Phase Transitions in Optical Lattices

We have observed Bragg scattering of photons from quantum degenerate ${}^{87}\mathrm{Rb}$ atoms in a three-dimensional optical lattice. Bragg scattered light directly probes the microscopic crystal structure and atomic wave function whose position and momentum width is Heisenberg limited. The spatial coherence of the wave function leads to revivals in the Bragg scattered light due to the atomic Talbot effect. The decay of revivals across the superfluid to Mott insulator transition indicates the loss of superfluid coherence.

Figure 1

Bragg scattering as a probe of the spatial wave function width. Bragg scattered intensity vs lattice depth in units of the recoil energy. The right axis gives the corresponding root-mean-square width of the wave function squared. The solid line is a no-free-parameter curve given by the Debye-Waller factor. Error bars are statistical. The inset is a schematic of the setup for Bragg scattering.

Figure 2

Bragg scattering as a probe of the momentum wave function width. Bragg scattered intensity vs the free-expansion time of the atoms after rapidly turning off the lattices from $15E_{\mathrm{R}}$. The decay in signal indicates a melting of the crystal structure or, in other words, spreading of the atomic wave packet with a given momentum uncertainty. The gray area is a no-free-parameter curve using the Debye-Waller factor taking into account the probe pulse duration of $5\mu \mathrm{s}$. Error bars are statistical.

Figure 3

Revivals of Bragg scattered light. Bragg scattered intensity vs the expansion time in three dimensions for three different initial lattice depths: $5E_{\mathrm{R}}$ (circles), $8E_{\mathrm{R}}$ (triangles), and $15E_{\mathrm{R}}$ (squares). Each data set is normalized to the Bragg intensity at the initial time. Revivals at lower lattice depths indicate coherence of the atoms across multiple lattice sites. The lines are phenomenological fits to exponentially decaying sine waves. Different lattice depth data are offset for clarity, and the error bar is representative. The inset is a solution to the one-dimensional Gross-Pitaevskii equation with experimental parameters and an initial state of a chain of Gaussian wave functions, showing the revivals of the density distribution as a function of time. The white dotted line represents the revival time.

Figure 4

Bragg scattering revival as a probe of superfluid coherence. Top: Bragg scattered revival signal at $60\mu \mathrm{s}$ vs lattice depth. Bottom: The top plot normalized to the Bragg scattered signal without expansion. We see a decrease in revival, and consequently superfluid coherence, as the lattice is increased. Error bars are statistical.