Coupling Identical one-dimensional Many-Body Localized Systems

We experimentally study the effects of coupling one-dimensional many-body localized systems with identical disorder. Using a gas of ultracold fermions in an optical lattice, we artificially prepare an initial charge density wave in an array of 1D tubes with quasirandom on-site disorder and monitor the subsequent dynamics over several thousand tunneling times. We find a strikingly different behavior between many-body localization and Anderson localization. While the noninteracting Anderson case remains localized, in the interacting case any coupling between the tubes leads to a delocalization of the entire system.

Figure 1

Coupling identical MBL systems: A charge density wave (CDW) with atoms only occupying even sites ($e$) is prepared in each of the identically disordered 1D tubes along the longitudinal ($x$) direction, with hopping $J$, on-site interaction energy $U$, and disorder strength $\mathrm{\Delta }$. Red and blue spheres indicate a typical distribution of $|\uparrow ⟩$ and $|\downarrow ⟩$ atoms. We monitor the time evolution of such a state for different intertube coupling strengths $J_{\perp }$, that is, different hopping amplitudes along the transverse ($y$) direction.

Figure 2

Time evolution of a charge-density wave (CDW): An initially prepared 1D CDW evolves in a disordered system with disorder strength $\mathrm{\Delta }=5J$ along one axis. We measure the imbalance $\mathcal{I}$ after a given evolution time in $1{\mathrm{D}}^{+}$ (circles), where $J_{\perp }\lesssim 10^{-3}J$ and in 2D (diamonds), where $J_{\perp }=J$. Each data point is the average of 6 disorder phase realizations, with error bars showing the standard error of the mean. Solid lines are fits [28] from which we extract the imbalance lifetimes. Shown in gray is an exact diagonalization (ED) calculation for $J_{\perp }=0$, $\mathrm{\Delta }=5J$, and $U=0$ [28].

Figure 3

Imbalance lifetimes vs interactions ($U$): Imbalance lifetimes at $\mathrm{\Delta }=5J$ for $1{\mathrm{D}}^{+}$ and 2D. We note that the $1{\mathrm{D}}^{+}$ case differs crucially from the ideal isolated 1D case. The lifetimes were extracted from fits to time traces such as in Fig. 2. Error bars denote fit uncertainty [28]. The gray shaded area indicates the range of measured atom number lifetimes, while the lines are guides to the eye.

Figure 4

Imbalance lifetimes in the 1D-2D crossover: The coupling strength $J_{\perp }$ between adjacent tubes is varied continuously for $\mathrm{\Delta }=5J$ at four different interactions $U$. Here, $J_{\perp }/J\lesssim 10^{-3}$ and $J_{\perp }/J={10}^0$ correspond to the $1{\mathrm{D}}^{+}$ and the 2D cases, respectively. The lack of saturation as $J_{\perp }\to 0$ indicates that the residual intertube coupling still limits the imbalance lifetime in the $1{\mathrm{D}}^{+}$ case. Solid lines denote power-law fits $\propto J_{\perp }^k$ for small $J_{\perp }$, with fitted exponents $k(U/J)$ of $k(0)=0.00(4)$, $k(-2)=-0.09(2)$, $k(-6)=-0.30(1)$, and $k(-20)=-0.16(3)$. We note that in principle the tunneling along the $z$ direction becomes sizable for the smallest $J_{\perp }$, as $J_{\perp }^z/J\sim 10^{-3}$.

Figure 5

Imbalance vs interactions ($U$) at fixed times: Data taken at $\mathrm{\Delta }=5J$ averaged over three different times ($38\tau -41\tau$) and 4 disorder phases $\varphi$. In the $1{\mathrm{D}}^{+}$ case (circles), the imbalance corresponds to the stationary value [14], whereas in the 2D case (diamonds) it is indicative of the decay lifetime. Solid lines are guides to the eye. The 2D case is limited to $U/J\le 10$ due to the details of the used Feshbach resonance [27].