Fourier relationship between the angle and angular momentum of entangled photons

We study the Fourier relationship between angle and orbital angular momentum of entangled photons. Spatial light modulators allow us to define and control the spatial mode measurement state. We observe strong quantum correlations, establishing that angular position and momentum distributions between the photons are related as conjugate Fourier pairs.

Figure 1

(Color online) Experimental apparatus. A frequency tripled Nd:yttrium-aluminum-garnet (Nd:YAG) laser at $355\mathrm{nm}$ is incident on a beta barium borate (BBO) crystal producing photon pairs at $710\mathrm{nm}$. Signal and idler photons are coupled using single mode fiber to avalanche photodiodes. Both the single channel and coincidence count rates are recorded as functions of the hologram design displayed on the spatial light modulator(s).

Figure 2

(Color online) Coincidence counts (red; dark gray in printed version) and single channel counts (gray) in channel $\left(A\right)$ as functions of ${\mathcal{l}}_A$ for the case that no aperture is implemented on SLM $B$. Note that the distribution of detected $\mathcal{l}$ states for the single channel $\left(A\right)$ is broad, but the coincidence count rate is only substantial for ${\mathcal{l}}_A=0$. The single channel count rate in channel $\left(B\right)$ is independent of ${\mathcal{l}}_A=0$ (not shown).

Figure 3

(Color online) Coincidence counts (red; dark gray in printed version) and single channel counts (gray) in channel $\left(A\right)$ as functions of ${\mathcal{l}}_A$ for the case that a twofold rotationally symmetric intensity aperture $(m=2)$ is implemented on SLM $B$. Note that the distribution of detected $\mathcal{l}$ states for the single channel is broad, but the coincidence count shows distinct sidebands for ${\mathcal{l}}_A=\pm 2$.

Figure 4

(Color online) As in Fig. 3, but for a twofold rotationally symmetric phase aperture implemented on SLM $B$. The coincidence count shows distinct sidebands for ${\mathcal{l}}_A=\pm 2$ and a suppression of the count rate at ${\mathcal{l}}_A=0$.