Realization of the Einstein-Podolsky-Rosen Paradox Using Radial Position and Radial Momentum Variables

As is well known, angular position and orbital angular momentum (OAM) of photons are a conjugate pair of variables that have been extensively explored for quantum information science and technology. In contrast, the radial degrees of freedom remain relatively unexplored. Here we exploit the radial variables, i.e., radial position and radial momentum, to demonstrate Einstein-Podolsky-Rosen correlations between down-converted photons. In our experiment, we prepare various annular apertures to define the radial positions and use eigenmode projection to measure the radial momenta. The resulting correlations are found to violate the Heisenberg-like uncertainty principle for independent particles, thus manifesting the entangled feature in the radial structure of two-photon wave functions. Our work suggests that, in parallel with angular position and OAM, the radial position and radial momentum can offer a new platform for a fundamental test of quantum mechanics and for novel application of quantum information.

Figure 1

Radial correlation of signal and idler photons in SPDC. (a) and (b) The radial positions are well correlated, $r_A=r_B$, while [(c) and (d)] the radial momenta are well anticorrelated, $p_A^r=-p_B^r$.

Figure 2

Experimental setup (see the text for more details).

Figure 3

Experimental results for radial position correlations. (a) Coincidence counts per second for $r_A$ and $r_B$ coordinates. (b), (c), and (d) The normalized probability corresponding to the lines across the regions in (a) where (I) $r_A=0.2$, (II) 0.3, and (III) 0.4 mm, respectively. Dots are experimental data, red solid lines are Gaussian fits, and vertical dashed lines indicate the central radial positions of $r_B=0.2$, 0.3, and 0.4 mm, respectively.

Figure 4

Experimental results for radial momentum correlations. (a) Coincidence counts per second for $p_A^r$ and $p_B^r$ coordinates. (b), (c), and (d) The normalized probability curves corresponding to the lines across the regions in (a) where (I) $p_A^r=-9$, (II) 0, and (III) $9\hslash {\mathrm{mm}}^{-1}$, respectively. Dots are experimental data, red solid lines are Gaussian fits, and vertical dashed lines indicate the central radial momenta of $p_B^r=9$, 0, and $-9\hslash {\mathrm{mm}}^{-1}$, respectively.