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2 Fig. 9. Intensity diﬀerence variance σs−i normalized to the SNL ns + ni . Each point (white circle) corresponds to a single-shot measurement where the spatial ensemble statistics has been performed over a 100 × 40 pixel region. The triangles (each one obtained by averaging the experimental points corresponding to a certain gain) and their linear ﬁt illustrate the trend of the data in the region between ns + ni = 8 and 20. 35) 2 Spatial Entanglement in Optical Parametric Down-Conversion 35 2 2 4 ys (e) hif t 4 6 (pi 6 8 xe ls) 8 10 10 if x sh ls) ix e t (p Fig.

Jedrkiewicz, and L. A. c. 1) where kj = nj ωj /c is the wave number of wave j at the carrier frequency along the z-axis; for an extraordinary wave the refractive index nj depends on the propagation direction, a property leading to spatial walk-oﬀ. In a single-pass conﬁguration with crystal length on the order of a few millimeters, the pump depletion due to down-conversion is indeed a small entity, unless extremely high intensity laser sources are used. We shall therefore work within the parametric approximation, which treats the pump as a known classical ﬁeld that propagates linearly inside the crystal, while the down-converted ﬁelds are quantized.

We can notice how the FWHM of the curves increases for 36 A. Gatti, E. Brambilla, O. Jedrkiewicz, and L. A. Lugiato increasing gain, clearly reﬂecting the increment of the speckle size (and thus of the transverse mode size) already observed in Fig. 7. , within one pixel) of the center of symmetry between the signal and the idler regions. It is interesting to note that the quantum nature of the correlation can also be estimated from the peak value of the correlation degree. 39) as also conﬁrmed experimentally within a good approximation.