Full-Field Quantum Correlations of Spatially Entangled Photons

Spatially entangled twin photons allow the study of high-dimensional entanglement, and the Laguerre-Gauss modes are the most commonly used basis to discretize the single-photon mode spaces. In this basis, to date only the azimuthal degree of freedom has been investigated experimentally due to its fundamental and experimental simplicity. We show that the full spatial entanglement is indeed accessible experimentally; i.e., we have found practicable radial detection modes with negligible cross correlations. This allows us to demonstrate hybrid azimuthal-radial quantum correlations in a Hilbert space with more than 100 dimensions per photon.

Figure 1

Schematic experimental setup. The spatially entangled photons are produced in the nonlinear crystal, whose surface is imaged with $7.5\times$ magnification ($f_{L1}=10\mathrm{cm}$, $f_{L2}=75\mathrm{cm}$) onto the spatial light modulator (display size $16\times 12{\mathrm{mm}}^2$, $800\times 600\mathrm{pixel}$). This device is programmed to perform the phase modulation required to transform the detection mode into the fundamental mode. The far field of the SLM surface is imaged ($10\times$, 0.2 numerical aperture objectives) onto the single-mode fiber which is connected to a single-photon counter. Simultaneous detection events from an entangled photon pair are selected in a coincidence time window of 2.3 ns. The inset shows an exemplary SLM phase pattern (${\ell }_s=1$, $p_s=1$; ${\ell }_i=-2$, $p_i=3$) and superimposed the magnification-corrected waist of the pump beam (purple), the waist of the detection single-mode fiber (1275 mm, yellow), and the detection-mode waist (red) $w=1000\mu \mathrm{m}$.

Figure 2

Quantum correlations between radial modes with different $p$ (for ${\ell }_s={\ell }_i=0$): Shown are the normalized (divided by maximum) coincidence count rates (color coded) as a function of the radial-mode numbers $p_s$ (horizontal axis) and $p_i$ (vertical axis) of the detection modes. Different rows depict results for different detection-mode waists as indicated. Left column, experimental data; right column: theoretical prediction. It is clearly visible that the smaller the detection-mode waist gets, the smaller the off-diagonal counts will be. This is a sign that we approach the Schmidt basis for $\gamma \to {\gamma }^{*}$. The detection-mode waists corresponds to waist ratios of (from top to bottom) $\gamma =2.4$, 3, 4.9; see 15.

Figure 3

The influence of the detection-mode waist: The quantity on the $y$ axis is a measure of how sharply the radial quantum correlations peak around $p_s=p_i$. For ideal LG mode detectors, we have $W\to 1$ for the waist ratio $\gamma \to {\gamma }^{*}$. For small waists ($w<500\mu \mathrm{m}$), pixilation effects are non-negligible. For larger waists, the agreement between experiment (circles) and theory (gray dots) is very good. The top axis indicates the ratio $\gamma$ of the pump-beam waist $w_p$ to the detection-mode waist $w=w_{s,i}$ of the signal and idler photon.

Figure 4

Quantum correlations between radial modes with different $p$, for given $\ell$ (rows) at a fixed waist ratio $\gamma =2.4$. The left column shows experimental data, the right column the theoretical prediction. The number of single-particle modes involved in these experiments is approximately 100.