Strongly Interacting Bosons in a Disordered Optical Lattice

We experimentally probe the properties of the disordered Bose-Hubbard model using an atomic Bose-Einstein condensate trapped in a 3D disordered optical lattice. Controllable disorder is introduced using a fine-grained optical speckle field with features comparable in size to the lattice spacing along every lattice direction. A precision measurement of the disordering potential is used to compute the single-particle parameters of the system. To constrain theories of the disordered Bose Hubbard model, we have measured the change in condensate fraction as a function of disorder strength for several different ratios of tunneling to interaction energy. We observe disorder-induced, reversible suppression of condensate fraction for superfluid and coexisting superfluid–Mott-insulator phases.

Figure 1

Fine-grained disorder superimposed on a 3D periodic optical lattice potential. (a) A polycarbonate holographic diffuser (gray disk) is used to create a 532 nm optical speckle field. The light (green) is focused by a 0.9 numerical aperture lens 13 mm from the atoms into the rectangular glass cell, where a BEC (blue circle) is confined in a three-dimensional optical lattice formed from three pairs of counterpropagating 812 nm laser beams (red). (b) The random optical potential (green) created by the optical speckle field adds independently to the lattice potential (red) to create a disordered potential. (c) A sample of the volume of the speckle intensity distribution (shown in false color). (d) The measured speckle intensity is used to reconstruct the AC function (false color). The $1/e^2$ radii of the AC distribution are 570 nm and $3\mu \mathrm{m}$ along the transverse and speckle propagation directions; the lattice axes project onto the $x$ and $z$ directions in this plane.

Figure 2

Probability density $\rho$ of the DBH parameters for $s=14$ and $\Delta =1$. The distribution for $\epsilon$ shown in (a) is one-sided, since the blue-detuned speckle potential can only increase the potential energy of a lattice site. In (b), the tunneling energy distributions are computed separately along the $x$ (blue) and $z$ (green) directions. The average of the ratio of tunneling energy along the $x$ and $z$ direction does not deviate by more than 10% even for moderately strong disorder ($\Delta \sim U$). The disorder does not shift the most probable values of $t$ and $U$, even though the mean values of these parameters change with increasing $\Delta$. The $\Delta =0$ values are $\epsilon =-10.85E_R$, $t=0.0095E_R$, and $U=0.360E_R$.

Figure 3

Measured condensate fraction $N_0/N$ in the disordered lattice potential. Data are shown for $s=(\mathrm{a})6$, (b) 12, and (c) 14. The disorder strength normalized to $U$ corresponds to $\Delta =0–7E_R$ for (a) and (b) and $\Delta =0–6E_R$ for (c). The hollow (solid) points are the condensate fraction after slow (fast) release from the lattice. Each data point is the average of more than six measurements at fixed $\Delta$ and $s$; the error bars show the statistical uncertainty. The inset to (c) is an image with a $400\mu \mathrm{m}$ field-of-view at $s=14$ and $\Delta =2E_R$ ($\Delta /U=5.7$). The cross-sectional profile of this image shows the fit to a two-component distribution (solid line) used to determine $N_0/N$. We used an empirical imaging noise model to determine that the systematic error in measurements of low $N_0/N$ was less than 0.03.

Figure 4

The reversible change in condensate fraction induced by disorder. Data are shown for $s=6$ (green triangles), $s=12$ (red squares), and $s=14$ (blue circles). Each data point is derived from a pair of slow and fast release points from Fig. 3; the errors bars are computed using the statistical uncertainties from Fig. 3. An effect unrelated to disorder is the reduction in $N_0/N$ for $s=14$ and $\Delta =0$, which reflects the presence of atoms in the MI phase in the pure lattice. The inset is an expanded view of the low $\Delta /U$ data for $s=14$.