Statistical properties of two-photon speckles

We experimentally study the statistics of two-photon speckle patterns, obtained when spatially entangled photon pairs are scattered through a random medium. Striking differences arise between the scattering of highly entangled states and almost separable states. Both the purity of the field and the Schmidt number, which quantifies the number of entangled modes, can be obtained from the visibility of the speckles. We observe nonexponential statistics for both the intensities and the two-photon correlations.

Figure 1

Entangled photon pairs are obtained via type I spontaneous parametric down conversion by pumping a 5-mm-thick periodically poled potassium titanium oxide phosphate (KTP) crystal with a laser beam (${\lambda }_p=413.1$ nm and 200 mW power). The crystal center is imaged onto the scatterer with two $f_1=200$ mm lenses. The far field of the scatter is imaged with a $f_d=250$ mm lens onto the detection plane. Detection occurs via projection onto two single-mode fibers. The size of the detection modes ($w_{\det }=140$ $\mu$m) determines the spatial resolution in the far-field plane. Narrow band spectral filters (5 nm at 826.2 nm) are used to selected down-converted light close to frequency degeneracy. The inset shows our scattering medium, which comprises two light shaping diffusers positioned in each other's far field (using a lens $f_c=10$ mm). This configuration mimics a volume scatter [18], but also allows sufficient counts to be measured. (a) Generation of a state with $K_{\mathrm{th}}=80$. The pump is weakly focused to a waist $w_p=160$ $\mu$m. (b) An almost separable state can be generated by focusing the pump to a spot that minimizes $K$. By using the relation between $K$, $w_p$, and $L$ from Ref. [21], we obtain $K_{\mathrm{th}}=1.4$ with a spot $w_p=11.5$ $\mu$m. This is done by removing the pump lens (not shown) and inserting a $f_2=100$ mm lens to focus the beam at the center of the crystal. The two $f_1$ lenses are removed and a single $f_3=59$ mm lens now images the center of the crystal on the scatter.

Figure 2

Coincidence counts measured by two scanning detectors for a single realization of the scattering medium. (a) A nonlocal speckle pattern, corresponding to the scattering of a highly entangled state with $K_{\mathrm{th}}=80$. (b) Separable speckle pattern, corresponding to the scattering of a state with a small number of modes, $K_{\mathrm{th}}=1.4$.

Figure 3

Probability distributions measured for $K_{\mathrm{th}}=1.4$. The (red) dashed curves are the theoretical distributions and the (black) dashed lines correspond to an exponential distribution. The insets show the results on a linear scale. (a) Distribution $P_1$ of intensities. (b) Distribution $P_2$ of coincidences for $x_1\ne x_2.$ (c) Distribution $P_2$ of coincidences for $x_1=x_2$.

Figure 4

Probability distributions measured for $K_{\mathrm{th}}=80$. The (red) curves are the theoretical distributions and the (black) dashed lines correspond to an exponential distribution. (a) Distribution $P_2$ of coincidences for $x_1\ne x_2$. (b) Distribution $P_1$ of intensities.