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Simulation of Anderson localization in two-dimensional ultracold gases for pointlike disorder

W. Morong and B. DeMarco
Phys. Rev. A 92, 023625 – Published 17 August 2015
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  • INTRODUCTION
  • PERCOLATION IN SPECKLE AND POINTLIKE…
  • LOCALIZATION LENGTHS: THE BORN…
  • BEYOND THE BORN APPROXIMATION
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Anderson localization has been observed for a variety of media, including ultracold atomic gases with speckle disorder in one and three dimensions. However, observation of Anderson localization in a two-dimensional geometry for ultracold gases has been elusive. We show that a cause of this difficulty is the relatively high percolation threshold of a speckle potential in two dimensions, resulting in strong classical localization. We propose a realistic pointlike disorder potential that circumvents this percolation limit with localization lengths that are experimentally observable. The percolation threshold is evaluated for experimentally realistic parameters, and a regime of negligible classical trapping is identified. Localization lengths are determined via scaling theory, using both exact scattering cross sections and the Born approximation, and by direct simulation of the time-dependent Schrödinger equation. We show that the Born approximation can underestimate the localization length by four orders of magnitude at low energies, while exact cross sections and scaling theory provide an upper bound. Achievable experimental parameters for observing localization in this system are proposed.

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    • Received 20 May 2015

    DOI:https://doi.org/10.1103/PhysRevA.92.023625

    ©2015 American Physical Society

    Authors & Affiliations

    W. Morong and B. DeMarco

    • Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801, USA

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    Vol. 92, Iss. 2 — August 2015

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    Images

    • Figure 1
      Figure 1

      Schematic representation of experimental implementation of pointlike disorder. Ultracold atoms are confined to a quasi-two-dimensional geometry using a sheet of far-detuned light (not shown). The disordered potential is generated by an additional laser beam (green shaded region) that passes through a holographic optic (disc) and is focused on the atoms (red spheres). The atoms experience a disorder potential consisting of a random arrangement of Gaussian barriers.

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    • Figure 2
      Figure 2

      Comparison of percolation in (a) a speckle potential and (b) a sparse pointlike disordered potential (b). Here, images of disorder potentials are shown in a scale-free fashion. The disorder potential energy is shown in false color. The color bar shows the potential energy in units of Δ. Regions in grayscale correspond to energies that are less than 30% the average disorder energy. A classical particle with this energy would be trapped in a finite-size region in the speckle potential, but able to propagate indefinitely for the pointlike disorder case.

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    • Figure 3
      Figure 3

      Percolation threshold for pointlike (blue squares) and speckle (red circles) disorder potentials. The dashed line is the known percolation threshold for speckle disorder, and the points are results of our simulation. For pointlike disorder, n and w are independently varied, and for speckle disorder, the correlation length was changed. Inset: The onset of significant percolation for pointlike disorder. The minimum detectable threshold level is 5×10−5 in our simulation. The error bars show the standard error of the mean for the average taken over 50 disorder realizations at each pointlike disorder point and eight realizations for each speckle disorder case.

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    • Figure 4
      Figure 4

      Localization lengths according to scaling theory within the Born approximation. Contour lines for three localization lengths are displayed using the experimental parameters described in the main text. The percolation threshold in terms of εk is shown using solid circles. If εk<Eth, then Al will not be observable and the atoms will be classically trapped. Because the average disorder potential energy is fixed at kB×1000 nK, V0 varies according to V0=kB×1989/n[μm−2] nK.

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    • Figure 5
      Figure 5

      Process of extracting scattering properties from a simulating propagation of a plane wave for the time-dependent Schrödinger equation. The initial wave vector k⃗ is in the y direction. (a) Real part of the wave function ψ(x⃗,τ) shown in false color. For all of the data shown in this figure, εk=kB×240 nK, V0=kB×1000 nK, and w=400 nm. Violation of the Born approximation is evident as significant distortion of the initial plane wave by the potential barrier at the origin (not pictured). (b) Real part of the scattered wave function ψsc(x⃗) obtained by subtracting a plane wave propagated in free space from the data shown in (a). The color bar applies to the simulated wave functions shown in (a) and (b). (c) The scattered probability density ψsc(x⃗)2 is shown as a density plot. The cross-hatched region is sampled to determine that differential cross section. (d) Differential cross section obtained from the data in (c). Points are shown at fixed θ and different r within the circular annulus.

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    • Figure 6
      Figure 6

      Ratio of total-collision cross section determined from the Born approximation σ(Bn) to σ calculated using the exact differential cross section for w=400 nm and V0=kB×50 nK. The error bars show the spread in values determined across the inner and outer radii of the annulus displayed in Fig. 5 and include the effects of numerical errors and deviation from the far-field limit.

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    • Figure 7
      Figure 7

      Localization lengths predicted using scaling theory and the exact differential cross section (circles) and the Born approximation (solid line). The density for the disorder potential is n=0.2μm−2, the average disorder potential energy Δ=kB×1000 nK, and the Gaussian barrier height V0=kB×9950 nK. The error bars are determined using the spread in the differential cross section determined across the inner and outer radii of the annulus displayed in Fig. 5.

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    • Figure 8
      Figure 8

      Simulated probability density distributions shown in false color from a time-dependent simulation of localization. A Gaussian wave packet initialized at t=0 in the disorder-free region marked by the white dashed line is shown in (a). The wave function propagated forward in time for 146 ms is shown in (b). The potential barriers that constitute the disorder potential are magnified and marked in black for clarity. For these simulations, εk=kB×25 nK, V0=kB×1000 nK, w=400 nm, and n=0.08μm−2.

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    • Figure 9
      Figure 9

      Procedure to determine ξ from the time-dependent simulations of localization. (a) Sample radial probability density showing the approach to a steady-state profile. The propagation times are τ=3 ms (solid black line), 15 ms (red dashed line), 30 ms (blue dot-dashed line), 90 ms (green dash-double-dotted line), 120 ms (short dashed line), and 150 ms (purple dotted line). For the data shown in this figure, εk=kB×100 nK, V0=kB×1000nK, w=400 nm, and n=0.08μm−2. (b) Fitted exponential decay lengths ξτ from data such as those in (a) for εk=kB×100 nK (red triangles), 35 nK (blue circles), and 3 nK (black squares). Only a fifth of the points generated for each value of εk are shown. The uncertainty in the points is too small to be visible. Fits of these data to an exponential function (lines) are used to extract the asymptotic value of the localization length ξ.

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    • Figure 10
      Figure 10

      Comparison of localization lengths from a numerical simulation of the time-dependent Schrödinger equation (blue squares), from the perturbative correction to the independent scattering transport properties (red circles) and from the Born approximation (solid black line). For comparison, the de Broglie wavelength 2π/k is shown as a dotted line. The parameters used for this plot are V0=kB×1000 nK, w=400 nm, and n=0.08μm−2. The error bars for the circles show the impact of the difference in the differential cross section across the inner and outer radii for the annulus shown in Fig. 5 and the standard deviation in the energy (resulting from the Gaussian envelope of the wave packet) for the squares.

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    • Figure 11
      Figure 11

      Thermal density profile for V0=kB×9947 nK, w=400 nm, n=0.2μm−2, and T=10 nK. Here, N is the two-dimensional number density for 10 000 particles.

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