Generation of Entangled States of Qudits using Twin Photons

We report an experiment to generate entangled states of $\mathcal{D}$-dimensional quantum systems, qudits, by using transverse spatial correlations of two parametric down-converted photons. Apertures with $\mathcal{D}$ slits in the arms of the twin photons define the qudit space. By manipulating the pump beam correctly, the twin photons will pass only by symmetrically opposite slits, generating entangled states between these different paths. Experimental results for qudits with $\mathcal{D}=4$ and 8 are shown. We demonstrate that the generated states are entangled states.

Figure 1

Outline of the experimental setup. The inset shows the transverse profile of the pump beam in the $\mathcal{D}$-slit plane, used in the measurements.

Figure 2

$\mathcal{D}=4$. $D_2$ single counts (○) and $D_1$-$D_2$ coincidence counts (●) measured simultaneously with $D_1$ fixed behind the slit $l$ and $D_2$ scanning in the $x$ direction. From left to right the single count peaks are the slits $l^{\prime }=-\frac{3}{2},\dots ,+\frac{3}{2}$. $D_1$ is fixed behind the slit $l$ (a) $+\frac{3}{2}$, (b) $+\frac{1}{2}$, (c) $-\frac{1}{2}$, and (d) $-\frac{3}{2}$. (e) Histogram of probabilities for all basis states $\{|l{⟩}_1|l^{\prime }{⟩}_2\}$.

Figure 3

$\mathcal{D}=8$. $D_2$ single counts (○) and $D_1$-$D_2$ coincidence counts (●) measured simultaneously with $D_1$ fixed behind the slit $l$ and $D_2$ scanning in the $x$ direction. From left to right the single count peaks are the slits $l^{\prime }=-\frac{7}{2},\dots ,+\frac{7}{2}$. $D_1$ is fixed behind the slit $l$ (a) $+\frac{7}{2}$, (b) $+\frac{5}{2}$, (c) $+\frac{3}{2}$, (d) $+\frac{1}{2}$, (e) $-\frac{1}{2}$, (f) $-\frac{3}{2}$, (g) $-\frac{5}{2}$, and (h) $-\frac{7}{2}$.

Figure 4

Fourth-order interference patterns for two photons transmitted by four slits as a function of the $D_1$ position. (a) $D_2$ is kept fixed at $x_2=0\mu \mathrm{m}$. (b) $D_2$ is kept fixed at $x_2=300\mu \mathrm{m}$. The solid curves were obtained from Eq. (8) with one normalization parameter.