Spin-orbit hybrid entanglement of photons and quantum contextuality

We demonstrate electromagnetic quantum states of single photons and of correlated photon pairs exhibiting “hybrid” entanglement between spin and orbital angular momentum. These states are obtained from entangled photon pairs emitted by spontaneous parametric down conversion by employing a $q$ plate for coupling the spin and orbital degrees of freedom of a photon. Entanglement and contextual quantum behavior (that is also nonlocal, in the case of photon pairs) is demonstrated by the reported violation of the Clauser-Horne-Shimony-Holt inequality. In addition, a classical analog of the hybrid spin-orbit photonic entanglement is reported and discussed.

Figure 1

Setup used for the two quantum-regime experiments. A Nd:YAG laser with average power of 150 mW at 355 nm pumps a nonlinear crystal of $\beta$-barium borate (BBO) cut for degenerate type I noncollinear phase matching which emits OAM-entangled $H$-polarized photon pairs at 710 nm (see Ref. [19] for details). The photons of each pair are split in arms A and B, respectively. Legend of the main components (see also graphic symbol legend in the upper left inset): $f_1$, $f_2$, lenses for beam control; QP, $q$ plate; HWP, half-wave plate; PBS, polarizer; M, mirror; SLM A and SLM B, spatial light modulators; IF, interference filter for bandwidth definition; $100\times$, microscope objectives for fiber coupling; ${\mathrm{D}}_A$, ${\mathrm{D}}_B$, photon detectors. In the classical-regime experiment, the optical line is the same as arm A. (Top-right inset) Computer-generated hologram patterns displayed on the two SLMs in the three experiments.

Figure 2

The experimental coincidence counts as a function of orientation of the sector hologram for different values of polarization direction, for (a) heralded single photons, (b) photon pairs, and (c) coherent-states. Black dots respresent $\theta =0$, dark gray dots $\theta =\pi /4$, gray dots $\theta =2\pi /4$, and light gray dots $\theta =3\pi /4$. The solid lines are the best theoretical fit over the experimental data. The fringe contrast is about 90$\%$, which is much larger than 70.7$\%$, as required for Bell’s inequality verification. (d) Simulated intensity and polarization distribution patterns of the optical field for the beam emerging from the $q$ plate in the case of horizontal polarization input beam.

Figure 3

The CHSH $S$ value in a region where it is larger than the classical limit $2$. The choice of the variables appearing in Eq. (4) is the following: $\theta =0$, ${\theta }^{\prime }=\pi /4$, $\chi$ is the plot abscissa, ${\chi }^{\prime }=\chi +\pi /8$. The light gray, gray, and black dots correspond to the experimental data in the case of single-photon (a), photon-pairs (b), and classical-wave (c) SAM-OAM experiments, respectively. The dashed line is the quantum mechanical ideal prediction. In the two cases (a) and (b), at $\chi =\pi /16$, the CHSH inequality is violated respectively by 17 and 10 standard deviations. The classical case (c) is plotted for comparison