Nonlocality of high-dimensional two-photon orbital angular momentum states

We propose an interferometric method to investigate the nonlocality of high-dimensional two-photon orbital angular momentum states generated by spontaneous parametric down conversion. We incorporate two half-integer spiral phase plates and a variable-reflectivity output beam splitter into a Mach-Zehnder interferometer to build an orbital angular momentum analyzer. This setup enables testing the nonlocality of high-dimensional two-photon states by repeated use of the Clauser-Horne-Shimony-Holt inequality.

Figure 1

Schematic of the proposed experimental setup. The boxes ${\mathrm{MZ}}_a$ and ${\mathrm{MZ}}_b$ represent the Mach-Zehnder interferometers in the path of the photon $a$ and $b$, respectively. The thick gray lines ${\mathrm{F}}_{xi}(x=a,b;i=1,2)$, represent the single-mode optical fibers. Each of them is coupled with a detector ${\mathrm{D}}_{xi}$. Further details are given in the text.

Figure 2

Detailed scheme of the OAM analyzer in the path of the photon $x=a,b$. BS denotes a 50-50 beam splitter, and ${\mathrm{VBS}}_x$ a variable-reflectivity beam splitter. The two channels “1” and “2” of the interferometer are indicated. With SPP we denoted the two spiral phase plates $(\chi =\alpha ,\beta )$, and ${\mathrm{M}}_1,{\mathrm{M}}_2,{\mathrm{M}}_3$ represent three ordinary mirrors. The role of ${\mathrm{M}}_3$ is to ensure that the photon undergoes an even number of reflections (thus maintaining the spatial symmetry of the input wave function), whichever path it takes.

Figure 3

Schematic drawing of a spiral phase plate (SPP) with a step index (= phase shift per unit angle) $\mathcal{L}=h_s(n-n_0)∕\lambda$, where $h_s$ is the step height, $n$ and $n_0$ are the refractive indices of the SPP and the surrounding medium, respectively, and $\lambda$ is the wavelength of the incident light. In this paper we assume $\mathcal{L}=\mathcal{l}+1∕2,\{\ell =1,2,\dots \}$. The orientation angle $\alpha$ is indicated.