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Spatial entanglement patterns and Einstein-Podolsky-Rosen steering in Bose-Einstein condensates

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Science  27 Apr 2018:
Vol. 360, Issue 6387, pp. 409-413
DOI: 10.1126/science.aao1850
  • Fig. 1 Extracting entanglement from spatially separated regions of a BEC.

    (A) Experimental sequence. In step 1, we prepare a BEC in a spin-squeezed state on an atom chip. In step 2, the trapping potential is switched off and the BEC expands. In step 3, a Rabi rotation pulse is applied to select the spin component Embedded Image to be measured, followed by recording two high-resolution absorption images of the atomic density distributions in states Embedded Image and Embedded Image. (B) Illustration of the spin-squeezed state on a sphere (Wigner function, representing the quantum fluctuations of the spin) and definition of the axes Embedded Image used in the measurement of the entanglement and EPR steering criteria. (C) Single-shot absorption images of the atomic densities in Embedded Image and Embedded Image, showing an example of regions A and B used to define the collective spins Embedded Image and Embedded Image entering in the entanglement and EPR steering criteria.

  • Fig. 2 Spatial entanglement patterns in the atomic cloud.

    (A) Entanglement criterion Eq. 1 evaluated for a spin-squeezed BEC (green points) for different horizontal positions of the one-pixel gap between regions A and B (see Fig. 1C), corresponding to different splitting ratios Embedded Image. Lines are a guide to the eye, and error bars indicate 1 SEM. The blue points show the maximum violation (minimum value of εEnt) that could be explained by detection cross-talk. (B) Entanglement between regions of different shapes (A = yellow, B = red) in a spin-squeezed BEC. The pixel pattern used for the analysis is illustrated above the respective data points, and the blue segments show the corresponding maximum violation expected by cross-talk.

  • Fig. 3 Observation of Einstein-Podolsky-Rosen steering.

    (A) EPR steering criterion Eq. 2, evaluated for steering AB (green filled circles) and BA (red filled circles) in a spin-squeezed BEC, for different horizontal positions of the one-pixel gap (see Fig. 1C), corresponding to different splitting ratios Embedded Image. EPR steering is strongest for intermediate splitting ratios. Empty circles: spin uncertainty relation involving the product of noninferred variances in region B (green) and A (red). Lines are a guide to the eye, and the shaded regions are the reduction of the uncertainty product in replacing the noninferred variances with the inferred ones. Blue points: maximum violation that could be explained by detection cross-talk. (B) EPR steering AB for different widths of the gap in Fig. 1C. The center of the gap is fixed to the position showing maximum EPR steering in Fig. 3A. Even for increased gap size, we find a significant violation of the bound, confirming that the correlations cannot be explained by detection cross-talk between the regions. Lines and shaded regions as in (A). (C) Atom number in regions A and B as a function of the gap size.

Supplementary Materials

  • Spatial entanglement patterns and Einstein-Podolsky-Rosen steering in Bose-Einstein condensates

    Matteo Fadel, Tilman Zibold, Boris Décamps, Philipp Treutlein

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    • Materials and Methods
    • Figs. S1 to S4
    • References
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