How to Observe High-Dimensional Two-Photon Entanglement with Only Two Detectors

We propose a novel setup to investigate the entanglement of orbital angular momentum states living in a high-dimensional Hilbert space. We incorporate noninteger spiral phase plates in spatial analyzers, enabling us to use only two detectors. The two-photon states that are produced are not confined to a $2\times 2$-dimensional Hilbert space, and the setup allows the probing of correlations in a high-dimensional space. For the special case of half-integer spiral phase plates, we predict that the Clauser-Horne-Shimony-Holt-Bell parameter $S$ is larger than achievable for two qubits ($S=2\sqrt{2}$), namely, $S=3\frac{1}{5}$.

Figure 1

(a) Schematic drawing of a SPP. The device shifts the phase of an incident beam proportional to the azimuthal angle $\theta$. (b) A calculated far-field diffraction pattern of a fundamental Gaussian beam after propagating through an $\ell =3\frac{1}{2}$ plate positioned in its waist plane, showing that rotational symmetry is broken. Black and white denote low and high intensity, respectively.

Figure 2

Proposed experimental setup. A nonlinear crystal (NLC) splits a pump photon in a signal photon and an idler photon by the process of spontaneous parametric down-conversion. In each path, a SPP (${\mathrm{SPP}}_{s,i}$) is inserted with a single-mode fiber ${\mathrm{F}}_{s,i}$, together forming the analyzer. The coincidence count rate of detectors ${\mathrm{D}}_{s,i}$ is measured as a function of the SPP angular orientations ${\alpha }_{\mathrm{s}}$ and ${\alpha }_{\mathrm{i}}$.

Figure 3

The overlap [see Eq. (3)] between a noninteger OAM state and the identical state rotated over $\alpha$. When $\lambda =0$, the states are identical aside from a trivial phase shift, due to the vanishing edge dislocation. For half-integer OAM, i.e., $\lambda =\frac{1}{2}$, the two states are generally different, leading to a parabolic dependence of their overlap; the two states are orthogonal when $\alpha =\pi$.