Excitations of One-Dimensional Bose-Einstein Condensates in a Random Potential

We examine bosons hopping on a one-dimensional lattice in the presence of a random potential at zero temperature. Bogoliubov excitations of the Bose-Einstein condensate formed under such conditions are localized, with the localization length diverging at low frequency as $\ell (\omega )\sim 1/{\omega }^{\alpha }$. We show that the well-known result $\alpha =2$ applies only for sufficiently weak random potential. As the random potential is increased beyond a certain strength, $\alpha$ starts decreasing. At a critical strength of the potential, when the system of bosons is at the transition from a superfluid to an insulator, $\alpha =1$. This result is relevant for understanding the behavior of the atomic Bose-Einstein condensates in the presence of random potential, and of the disordered Josephson junction arrays.

Figure 1

An example of the correspondence between initial distributions and the exponent $g=\alpha$ at the end of the flow. Main plot: terminal $g$ as a function of the parameter $\delta$ for initial Gaussian coupling distributions $P_U(U)\sim e^{-(U-0.3{)}^2/{0.7}^2}$, and $P_J(J)\sim e^{-(J-\delta {)}^2/{0.4}^2}$, truncated for $J$, $U<{10}^{-4}$. The transition (terminal $g=1$) occurs at $\delta \approx 1.05$. Inset: example of the flow of $g$ vs the RG flow parameter $\Gamma$ for Gaussian initial conditions with $\delta =1.125$. The localization exponent $\alpha$ coincides with $g$ at the large $\Gamma$ plateau. Each point is averaged over 40 realizations of chains $5\times {10}^6$ long.