Observation of two-dimensional Anderson localisation of ultracold atoms

Abstract

Anderson localisation —the inhibition of wave propagation in disordered media— is a surprising interference phenomenon which is particularly intriguing in two-dimensional (2D) systems. While an ideal, non-interacting 2D system of infinite size is always localised, the localisation length-scale may be too large to be unambiguously observed in an experiment. In this sense, 2D is a marginal dimension between one-dimension, where all states are strongly localised, and three-dimensions, where a well-defined phase transition between localisation and delocalisation exists as the energy is increased. Here, we report the results of an experiment measuring the 2D transport of ultracold atoms between two reservoirs, which are connected by a channel containing pointlike disorder. The design overcomes many of the technical challenges that have hampered observation of localisation in previous works. We experimentally observe exponential localisation in a 2D ultracold atom system.

Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory

“…constituting what is known as "point-like" disorder, chosen for its lower percolation threshold [8]. This system could be experimentally realised with cold atoms as in [15], where an attractive 2D trap is used to contain atoms in a planar geometry, a repulsive custom potential generated by a spatial light modulator (SLM) allows the atoms to be confined to, for example, a rectangular box, and Gaussian point-like scatterers are generated by imaging squares of light produced by the SLM.…”

“…Lengths will be measured in units of , energy is units of E 0 = ħ h 2 /(2m 2 ), and time in t 0 = ħ h/E 0 . Typically, for a cold-atom experiment such as [15], ∼ 1 µm, E 0 ∼ 1 nK ×k B , and t 0 ∼ 5 ms.…”

“…In all the simulations to follow, L/ and W / are further chosen as integers. This restriction is imposed to stay in line with the discrete nature of the pixelated SLM used in [15] to both set the geometry and produce the scatterers. In the case of this experiment, one could reasonably choose to be the length of the side of the squares imaged on the SLM to produce the disorder.…”

“…Understandably, this method is inaccurate for strong disorder. Exact time-dependent simulations with the Schrödinger [6,8,13,14] or Gross-Pitaevskii [15,16] equations can be used instead, but this approach is very time-consuming and yields little insight into the physics. Finally, access to the localisation length directly through the eigenstates of the Hamiltonian is hampered by practical considerations (as we shall show below).…”

“…Out of the studies above, one-dimensional (1D) [6, 13, 17-19, 21-23, 30, 31, 34, 35, 37, 38] and two-dimensional (2D) [6-8, 10, 12, 14, 15, 20-28, 36] models have been numerically explored far more thoroughly than three-dimensional (3D) [6,27,34], simply because of the increased computational requirements of higher-dimensional spaces. Possibly the most heavily studied model of localisation is the Anderson model, also known as a tight-binding Hamiltonian [4, 10, 12, 13, 22-27, 29, 30, 32, 34-37, 39-44], but other examples include the kicked rotor [19] (formally equivalent to the Anderson model), the Lloyd model [13,21], the Peierls chain [38], a quantum walker [31], and the continuous Schrödinger equation [13,14,17], with either a speckle potential [7,16], delta-function point scatterers [6,10], or more realistic Gaussian scatterers [8,15].…”