Observation of Subdiffusion in a Disordered Interacting System

Abstract

We study the transport dynamics of matter-waves in the presence of disorder and nonlinearity. An atomic Bose-Einstein condensate that is localized in a quasiperiodic lattice in the absence of atom-atom interaction shows instead a slow expansion with a subdiffusive behavior when a controlled repulsive interaction is added. The measured features of the subdiffusion are compared to numerical simulations and a heuristic model. The observations confirm the nature of subdiffusion as interaction-assisted hopping between localized states and highlight a role of the spatial correlation of the disorder. [1-3, 7, 11, 12, 14-16, 18]. Most authors agree that the nonlinearity should prevent localization and that the wavepacket should expand in a way which is slower than normal diffusion. However, experimental evidence of such subdiffusive expansion is to date still missing.Here we study the dynamics of Bose-Einstein condensates with controllable nonlinearity expanding along a one-dimensional quasiperiodic lattice. Despite its large spatial correlation, this kind of potential is known [20,21] to feature exponentially localized states that are equivalent to those appearing in lattices with uncorrelated disorder described by the Anderson model [22]. Such a system has been successfully exploited for the investigation of the delocalizing effect of repulsive interactions on a trapped system in equilibrium [23,24]. If a noninteracting gas is let free to expand along the quasiperiodic lattice, no transport is observed [25] because all single-particle eigenstates are localized. By adding a controlled interatomic repulsion, we now observe a slow increase of the width σ of the sample that asymptotically follows a subdiffusive law: σ(t) ∝ t α , with α = 0.2 − 0.4. We find that the exponent increases with the interaction energy, in qualitative agreement with both numerical simulations based on a 1D discrete nonlinear Schrödinger equation (DNLSE) of the quasiperiodic lattice [7] and the predictions of a heuristic model. Our observation confirms the nature of subdiffusion as interaction-assisted hopping between localized states. The observed exponents are however larger than the one calculated for uncorrelated disordered potentials [1-3, 11, 12, 14-16, 18], suggesting a role of the spatial correlation of the disorder.The one-dimensional quasiperiodic potential is created by perturbing a primary optical lattice with a weaker incommensurate lattice [26]:V (x)=V 1 cos 2 (k 1 x) + V 2 cos 2 (k 2 x). Here k i =2π/λ i are the wavevectors of the lattices (λ 1 =1064.4 nm and λ 2 =859.6 nm). This potential is characterized by the spacing d=λ 1 /2 and the tunneling energy J of the primary lattice, and by the disorder strength ∆, which scales linearly with V 2 [27]. In the case of non-interacting particles this system constitutes an experimental realization of the Harper or AubryAndré model [20,21] which shows a transition between extended and localized states for a finite value of the disorder ∆/J = 2 [23,27]. Above this threshold all single-partic...

Inelastic Multiple Scattering of Interacting Bosons in Weak Random Potentials

“…All the real space experiments were explicitly done for non-interacting particles and, for 3D, in the regime of a very strong disorder potential and/or for very low single-particle energies (in agreement with k dis 1). The inclusion of inter-particle interaction rather leads to a suppression of Anderson localization [28][29][30] and, as demonstrated recently, to sub-diffusive behavior in 1D [31]. In 2D and 3D, however, the understanding is still far from complete, see [32][33][34] and references therein.…”

“…(31) indicated in Chapter 2, and finally apply the contact interaction approximation to eq. (31), that leads to an explicit expression forT (1) U (E 12 ). We again employ units in which h/2m ≡ 1.…”

“…Since, however, we assume the extension of the scattering region V in any spatial direction to be much larger than the disorder mean free path dis , we can safely ignore the associated small width and work with the T -matrix as stated in eq. (31).…”

“…Write down the corresponding two-body T -matrix element, see (31), for any collision between two particles (denoted by a solid and an empty square in Figs. 4a) and b), respectively).…”

Probing the effect of interaction in Anderson localization using linear photonic lattices

“…In fact, Anderson himself first noticed the importance of interaction in localization phenomena [8] and launched a theoretical investigation in collaboration with Fleishman [9]. Recently, advances in technology have reinvigorated theoretical [10,11] and experimental [12][13][14][15][16][17][18][19] interest on this subject. For the special case of two interacting particles in a random one-dimensional (1D) potential, Shepelyansky has investigated the interplay of disorder and interaction and concluded that interaction modifies (weakens) localization [20].…”