Glassy Behavior in a Binary Atomic Mixture

We experimentally study one-dimensional, lattice-modulated Bose gases in the presence of an uncorrelated disorder potential formed by localized impurity atoms, and compare to the case of correlated quasidisorder formed by an incommensurate lattice. While the effects of the two disorder realizations are comparable deeply in the strongly interacting regime, both showing signatures of Bose-glass formation, we find a dramatic difference near the superfluid-to-insulator transition. In this transition region, we observe that random, uncorrelated disorder leads to a shift of the critical lattice depth for the breakdown of transport as opposed to the case of correlated quasidisorder, where no such shift is seen. Our findings, which are consistent with recent predictions for interacting bosons in one dimension, illustrate the important role of correlations in disordered atomic systems.

Figure 1

Disordered one-dimensional Bose gases. (a) To create a disordered impurity field, half the atoms of a lattice-trapped Bose gas are converted to an auxiliary spin state and localized to a state-selective incommensurate lattice (dashed green). (b) Alternatively, a weak incommensurate lattice is superimposed onto a lattice-trapped Bose gas. In both cases, the secondary potential (atomic or optical) causes site-dependent energy shifts ${\epsilon }_i$ and site-to-site energy differences ${\Delta }_i$.

Figure 2

(a),(b) Histograms of calculated $\Delta$ distributions for the cases of the mixture (assuming either 0 or 1 impurities per site) and an incommensurate lattice of depth $s^{\prime }=1$. Both are for a primary lattice depth $s=6$ ($U/E_R=0.4$). (c),(d) The autocorrelation function ${\chi }_j=⟨{\Delta }_i{\Delta }_{i+j}{⟩}_i/⟨{\Delta }_i{\Delta }_i{⟩}_i$ of the $\Delta$ distributions in (a),(b), as a function of the site-to-site distance $j$. The averaging is over 1000 sites.

Figure 3

Disappearance of excitation gap due to disorder. (a) Visibility as a function of amplitude-modulation frequency (normalized to $U/E_R=0.53$ for $s=14$), in the absence of disorder for $s=9$ and $s=14$ (open and filled black circles). Fit lines are two Gaussians on a linear slope. (b) For $s=14$, with atomic impurities, open purple squares and filled red squares represent $f_{\mathrm{imp}}=0.1$ and 0.5, respectively. (c) For $s=14$, with an incommensurate lattice of depth $s^{\prime }=1$ (orange triangles) and no impurities. Solid error bars are statistical over several runs, while dashed are estimated errors for individual runs (120% of maximum statistical error).

Figure 4

(a) The response to impulse is determined by a straight-line fit (with slope $\alpha$, normalized to the case of free atoms) to the dependence of velocity on applied impulse $I$ (profiles shown for disorder-free and impurity mixture cases; recoil velocity $v_R=h/2md$). (b) $\alpha$ versus lattice depth $s$ for bosons without disorder (black circles) and with atomic impurities ($f_{\mathrm{imp}}=0.5$, red squares). Lines and surrounding shaded regions are fits to the data with confidence regions (1 s.d.) of a linearly decaying response, with no response ($\alpha =0$) beyond a depth $s_c$. (c) Similarly, but for an incommensurate lattice ($s^{\prime }=3$, green diamonds), with disorder-free data reproduced for comparison. The inset shows values of $s_c$ versus $s^{\prime }$ as determined by fit intercepts. (d) The momentum-peak width $\sigma$ ($1/\sqrt{e}$ half-width) is determined by a symmetric multi-Gaussian fit to time-of-flight interference patterns. The shaded regions about $v/v_R=0(\pm 1)$ are used to count $N_{0(\pm 1)}$ for visibility measurements of Fig. 3. (e) $\sigma$ versus $s$ for 1D bosons without disorder and for the mixture [colors and symbols as in (b)]. The straight line for the disorder-free data is a fit of a linear increase beyond a depth $s_c$. The first few data points for the case of atomic impurities are connected as a guide to the eye. (f) Similarly, but for an incommensurate lattice with $s^{\prime }=2$ (blue triangles) and with disorder-free data reproduced.