Selective state spectroscopy and multifractality in disordered Bose-Einstein condensates: a numerical study

Abstract

We propose to apply a modified version of the excitation scheme introduced by Volchkov et al. on bosons experiencing hyperfine state dependent disorder to address the critical state at the mobility edge of the Anderson localization transition, and to observe its intriguing multifractal structure. An optimally designed, spatially focused external radio frequency pulse can be applied to generate transitions to eigenstates in a narrow energy window close to the mobility edge, where critical scaling and multifractality emerge. Alternatively, two-photon laser scanning microscopy is proposed to address individual localized states even close to the transition. The projected image of the cloud is shown to inherit multifractality and to display universal density correlations. Interactions - unavoidably present - are taken into account by solving the Gross-Pitaevskii equations, and their destructive effect on the spectral resolution and the multifractal spectrum is analyzed. Time of flight images of the excited states are predicted to show interference fringes in the localized phase, while they allow one to map equal energy surfaces deep in the metallic phase.

Figure 1

(a) Projected image of a critical multifractal eigenstate of size (L/a)³ = 100³, exhibiting clustering. (b–d) Schematics of the proposed experiment: (b) The condensate is prepared in the hyperfine state |↓〉 where no disorder is present. The external field excites the atoms to the hyperfine state |↑〉, interacting with the disorder potential V_↑. The energy E_↑ of the final state is controlled by the frequency ω. (c) Excited atoms in the |↑〉 state are imaged by a horizontal laser beam. The overlap between different eigenstates can be reduced by using a narrow excitation beams of waist $w_0\lesssim \xi$, where ξ is the typical size of a localized state. (d) Even a single localized state can be excited and imaged by crossing two laser beams.

Figure 2

(a) Spectral weight (red line) w_↑(E_↑) of the ↑ bosons as a function of the final state’s energy E_↑ after the excitation pulse for a lattice of (L/a)³ = 40³ sites, W = 17J and excitation beam waist w₀ = 4a. We simulated N = 3200 atoms with typical interaction strengths U_↑↓ = U_↓↓ = 2J, excitation frequency ω ≈ 8J/ħ. The grey area indicates the smoothed density of states of the ↑ bosons in the self-consistent disorder potential that combines the static external disorder and the repulsion of the ↓ background. Inset: same spectral peak on a smaller scale. (b,c) Projected images of a metallic (W = 10J) and a localized (W = 28J) final state. The excitation beam has vertical direction in both cases. (d) Projected image of a single, slightly localized state (W = 17J), excited by two crossed laser beams. All figures show images of size 40a × 40a.

Figure 3

(a) Rescaled correlation function of the projected densities ${\hat{\rho }}_{\mathbf{x}\uparrow }$ for different disorder strengths but fixed excitation frequency ω ≈ 8J/ħ in the localized regime. Correlations were measured along the y direction in Fig. 1. Other parameters were set as in Fig. 2. Values of ξ(W) were determined by collapsing the curves. Inset: divergence of ξ at W → W_c ≈ 16.5J. (b) Multifractal spectrum extracted from the probability distribution of the projected density ${\hat{\rho }}_{\uparrow }$. The measured spectrum of non-interacting bosons excited by a short pulse (red circles) follows closely the projected spectrum of the critical state (thick green line). The location of the maximum, α_max ≈ 2.3, characterizes the anomalous scaling of the typical projected density, ${\hat{\rho }}_{\mathbf{x}\uparrow }^{\mathrm{typ}}\sim L^{-{\alpha }_{\max }}$. The multifractal spectrum gets distorted upon increasing interaction strength. (c) Spectrum of the final state for a fixed disorder configuration as a function of U/J. The sharp spectral peaks are shifted and broadened with increasing interaction, leading to eigenstate overlaps in the projected images and resulting in the degradation of the measured multifractal spectrum. Inset: average spectral peak width (full width at half maximum) as a function of U for N = 3200.

Figure 4

Time of flight image of excited atoms in the localized (left) and in the delocalized phase (right). The approximate self-consistent energies of the shown localized and delocalized states are E_↑ = 1.9 J and E_↑ = 0.6 J respectively. In the localized phase interference fringes emerge, while in the delocalized phase a “Fermi surface” structure tracing smeared equal energy surfaces in momentum space appears.