Einstein-Podolsky-Rosen Paradox in Twin Images

Spatially entangled twin photons provide both promising resources for modern quantum information protocols, because of the high dimensionality of transverse entanglement, and a test of the Einstein-Podolsky-Rosen paradox in its original form of position versus impulsion. Usually, photons in temporal coincidence are selected and their positions recorded, resulting in a priori assumptions on their spatiotemporal behavior. In this Letter, we record, on two separate electron-multiplying charge coupled devices cameras, twin images of the entire flux of spontaneous down-conversion. This ensures a strict equivalence between the subsystems corresponding to the detection of either position (image or near-field plane) or momentum (Fourier or far-field plane). We report the highest degree of paradox ever reported and show that this degree corresponds to the number of independent degrees of freedom, or resolution cells, of the images.

Figure 1

Experimental setups used to image correlations. (a) Measurement of momentum correlations with the cameras in the focal plane. Inserts: sums of 700 far-field images; $p{x}_1=-p{x}_2,p{y}_1=-p{y}_2$ are the coordinates of twin pixels. (b) Cameras in the crystal image plane and sums of 700 near-field images with twin pixels in $x_1=x_2,y_1=y_2$.

Figure 2

Joint probabilities versus the transverse spatial coordinates. Color scales are expressed in coincidence counts over 35 000 pairs of images, corrected from the mean corresponding to accidental coincidences: (a),(b) near field; (c),(d) far field. (e) In the far field, the correlations arise in a coherence area that is larger for momenta the most distant from the pump direction, in the direction $x$ along which the two fluorescence beams are separated from the walk-off.

Figure 3

Normalized cross-correlation functions in position and momentum: The cross-correlation is calculated over 700 images in the far field (a),(c) and image plane (b),(d). In (c) and (d) are presented cross-correlation of images that do not share any pump pulses.

Figure 4

Normalized cross-correlation function versus the number of images: left images show correlation computed on the physical pixels (only the central part is presented). Smaller right images show correlation computed after grouping $8\times 8$ pixels: (a),(b) far field; (c),(d) near field.