Localization in momentum space of ultracold atoms in incommensurate lattices

We characterize the disorder-induced localization in momentum space for ultracold atoms in one-dimensional incommensurate lattices, according to the dual Aubry-André model. For low disorder the system is localized in momentum space, and the momentum distribution exhibits time-periodic oscillations of the relative intensity of its components. The behavior of these oscillations is explained by means of a simple three-mode approximation. We predict their frequency and visibility by using typical parameters of feasible experiments. Above the transition the system diffuses in momentum space, and the oscillations vanish when averaged over different realizations, offering a clear signature of the transition.

Figure 1

Quasimomentum distribution $|f_k(t)|{}^2$ obtained from the DFT of the solution of Eq. (1). The index $\xi$ is related to $k$ by $\xi =2\pi k/N$. Here we use $\alpha =0.2282\dots$ and $\phi =0$. The initial wave packet in real space is a Gaussian of width $\sigma =10$. Time is given in dimensionless units. Top panel: $\lambda =1$; only a few modes are involved and a periodic oscillation of the central and side peaks is observed. The side peaks are at a distance $\pm 2\pi \alpha$ from the central peak. Bottom panel: $\lambda =5$; the evolution is affected by the coupling of many modes and the periodic oscillations are no more visible.

Figure 2

Oscillation frequency of the central peak of quasimomentum ($\lambda <2$, red) and spatial ($\lambda >2$, blue) distributions for $\sigma =10$, $\phi =0$. The full solution of the AA model (dots) is compared with the predictions of an analytic three-mode approximation (full lines) and a semianalytic five-mode approximation (dashed line).

Figure 3

Spatial distribution $\left|{\psi }_n\right|{}^2$ obtained from the solution of Eq. (1) using a single-site initial condition ${\psi }_n={\delta }_{n,0}$. As in Fig. 1, we use $\alpha =0.2282\dots$ and $\phi =0$. Time is given in dimensionless units. Top panel: $\lambda =1$; the initially localized wave packet spreads ballistically and there are no visible periodic oscillations. Bottom panel: $\lambda =5$; the wave packet is localized and a periodic oscillation of the central peak and its nearest neighbors is clearly visible.

Figure 4

Visibility of the oscillations in real and quasi-momentum space as a function of $\lambda$, for $\phi =0$. The numerical points are compared with the three-mode approximation (full lines). Bottom: Only a few modes have been initially populated by using a $\delta$-function initial wave packet in real space and a Gaussian with $\sigma =10$ in momentum space. Top: Many modes have been initially populated by inverting the initial conditions with respect to the bottom panel.

Figure 5

Phase-averaged intensity of the central peak in momentum space, $|f_0(t)|{}^2$, as a function of time and disorder strength $\lambda$ for a wave packet with $\sigma =10$. The intensity is given in arbitrary units.