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Effects of disorder upon transport and Anderson localization in a finite, two-dimensional Bose gas

Mojdeh S. Najafabadi, Daniel Schumayer, and David A. W. Hutchinson
Phys. Rev. A 104, 063311 – Published 13 December 2021
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  • INTRODUCTION
  • DETAILS OF SIMULATIONS
  • RESULTS
  • CONCLUSION
    • References

    Abstract

    Anderson localization in a two-dimensional ultracold Bose-gas has been demonstrated experimentally. Atoms were released within a dumbbell-shaped optical trap, where the channel of variable aspect ratio provided the only path for particles to travel between source and drain reservoirs. This channel can be populated with columnar (repulsive) optical potential spikes of square cross section with arbitrary pattern. These spikes constitute impurities, the scattering centers for the otherwise free propagation of the particles. This geometry does not allow for classical potential trapping which can be hard to exclude in other experimental setups. Here we add further theoretical evidence for Anderson localization in this system by comparing the transport processes within a regular and a random pattern of impurities. It is demonstrated that the transport within randomly distributed impurities is suppressed and the corresponding localization length becomes shorter than the channel length. However, if an equal density of impurities are distributed in a regular manner, the transport is only modestly disturbed. This observation corroborates the conclusions of the experimental observation: the localization is indeed attributed to the disorder. Beyond analyzing the density distribution and the localization length, we also calculate a quantum “impedance” exhibiting qualitatively different behavior for regular and random impurity patterns.

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    • Received 22 September 2021
    • Accepted 26 November 2021

    DOI:https://doi.org/10.1103/PhysRevA.104.063311

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Anderson localizationCold and ultracold moleculesLocalizationMany-body localizationScattering theoryWeak localization
    1. Techniques
    Anderson impurity model
    Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

    Authors & Affiliations

    Mojdeh S. Najafabadi*, Daniel Schumayer, and David A. W. Hutchinson†

    • Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Otago, Dunedin 9016, New Zealand

    • *shimo048@postgrad.otago.ac.nz
    • †david.hutchinson@otago.ac.nz

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    Vol. 104, Iss. 6 — December 2021

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    Images

    • Figure 1
      Figure 1

      On the left the schematic of the dumbbell potential is shown: L is the length of the channel, W is the width of the channel, and R is the common radius of the two circular reservoirs. On the right the single columnar potential spike is depicted, showing its square cross section of area σ and height/strength of Vimp.

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    • Figure 2
      Figure 2

      The impurity potential Vimp is depicted for randomly (a) and regularly (b) distributed potential spikes for the same fill-factor η=0.25. Only the channel segments of the dumbbell trap are shown.

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    • Figure 3
      Figure 3

      The left and right columns of graphs show ρ2D and ρ1D, respectively, at different moments in time, covering the entire duration of the simulation. In the right column, ρ1D is plotted on a logarithmic scale, while the left and right reservoirs are depicted as shaded areas. The channel length is 180 μm. While there is a small portion of the density falling outside of the dumbbell potential, it does not show up in these graphs, as this portion is negligible and thus several orders lower in magnitude than that within the dumbbell.

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    • Figure 4
      Figure 4

      In the right column the time evolution of ρ1D is depicted for η=0.05 (top) and η=0.2 (bottom) at t=2.49, 12.44, and 62.19 and at t=248.76 ms. In the left column ρ2D is shown with the contour of the dumbbell potential overlaid. The geometry is determined by (L,W,R)=(180,36,43) μm.

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    • Figure 5
      Figure 5

      Figure depicts the one-dimensional density profile over the entire numerical box for η=0 (a), 0.1 (b), and 0.2 (c). The vertical dashed lines separate the five important regions: the outermost regions are outside of the dumbbell trap, while the central three regions are the left reservoir, the channel, and the right reservoir. Within the channel the density profile is plotted using two colors for the left and right sides of the channel, respectively. The graph also shows the linear fits to ρ1D. The fitted localization lengths ξleft and ξright, with the corresponding goodness-of-fit measures, Rfit2, are given at above the central density.

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    • Figure 6
      Figure 6

      Localization lengths ξleft and ξright are depicted as functions of η with blue plus markers (+) and red crosses (×), respectively (left axis). The corresponding measure for goodness-of-fit Rfit2 is also shown with solid circles and applying the same color coding (right axis).

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    • Figure 7
      Figure 7

      The apparent localization length ξleft is drawn at each time step for η=0.02, 0.05, 0.1, and 0.2 (bottom four curves). The corresponding goodness-of-fit measures are also plotted with matching color (top four curves). The dumbbell geometry is given by (L,W,R)=(180,36,45) μm. The horizontal gray dashed lines represent the length of the channel, L, and its half, L/2, in order to provide comparison.

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    • Figure 8
      Figure 8

      The one-dimensional density in a short channel is plotted at the end of the time evolution. The left and right reservoirs are indicated with shaded areas and vertical thin blue dashed lines. The thicker dashed lines in the middle of the graph are the linear fits to ρ1D. The dumbbell geometry is (L,W,R)=(36,36,58) μm.

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    • Figure 9
      Figure 9

      Localization length ξ as a function of η for a dumbbell with a short channel length of 36 μm. The symbols and colors are the same as those in Fig. 6.

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    • Figure 10
      Figure 10

      Momentum distribution of atoms. Panel (a) depicts the initial distribution of the modulus of momentum after releasing the initial harmonic trap (t=1 ms). The distribution of the modulus of momentum is also plotted in the right well (b), in the channel (c), and in the left well (d) after t>250 ms. The curves in panels (b)–(d) correspond to three fill-factors: η=0.0 (black dash-dotted line), η=0.1 (blue solid line with star-shaped marker), and η=0.2 (red solid line with triangle marker). The dumbbell parameters are (L,W,R)=(180,36,45) μm.

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    • Figure 11
      Figure 11

      A comparison of two-dimensional (top panels) and one-dimensional densities (bottom panels) after t=250 ms time of expansion for impurities distributed regularly and randomly. The top panels provide a visual representation of ρ2D within the dumbbell and suggest qualitatively different behavior for the two cases. In the case of regularly distributed impurities the density seems to be more or less uniform although it also shows weak filamentary structures, while the randomly distributed impurities seem to result in a more localized density distribution. The triangular shape of ρ1D on the logarithmic scale is apparent for randomly distributed impurities, suggesting localization. The left and right reservoirs are indicated with shaded areas and vertical blue dashed lines.

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    • Figure 12
      Figure 12

      Normalized number of particles in the channel (two curves starting on the top at t=0) and in the reservoirs (two curves starting at the bottom at t=0) are depicted as functions of time.

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    • Figure 13
      Figure 13

      Momentum distributions of atoms are shown in the three segments of the dumbbell for regularly and randomly distributed impurities at fill-factor η=0.2. The left and right wells are depicted in panels (a) and (c), while panel (b) represents the channel segment. The black solid curves belong to the case of regularly distributed impurities, while the blue solid curves with star-shaped markers depict the data for the random distribution. The dumbbell geometry is given by (L,W,R)=(180,36,45) μm.

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    • Figure 14
      Figure 14

      The absolute value of impedance, |Z|, is plotted as a function of η for randomly (top curve) and regularly (bottom curve) distributed scatterers. The geometry is given by (L,W,R)=(180,36,45) μm.

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