Verification of continuous variable entanglement with undetected photons

Abstract

We verify transverse spatial entanglement of photon-pairs generated in spontaneous parametric down conversion using a nonlinear interferometric technique without relying on any coincidence detection. We experimentally demonstrate the violation of the Einstein-Podolsky-Rosen criterion and of the Mancini-Giovannetti-Vitali-Tombesi criterion using single photon interference of one of the photons of the pairs. We also provide a comprehensive theoretical analysis. The experimental results that we have obtained show good agreement with the theoretical values. Our method performs well under experimental losses and can be applied to highly non-degenerate sources, where there are no suitable detectors for one of the photons in the quantum state and our method could also be extended to the discrete degrees of freedom to certify high-dimensional (OAM) entanglement.

Figures (4)

• Figure 1: Comparison between the principle of standard coincidence-based entanglement measurement and our method using induced coherence. (a) Standard approach: Spatial entanglement of photon pairs generated via SPDC is typically measured by detecting both signal and idler photons in coincidence. Scanning detectors are usually used to record joint probability distributions in position or momentum bases. (b) Our approach: Using an induced coherence interferometer, only the signal photon is detected on an EMCCD camera, while the idler photon remains undetected. By analyzing the interferometric image of the knife-edge in position and momentum space, we retrieve the same conditional variances without coincidence detection.
• Figure 2: Experimental setup for measuring the edge-spread function (ESF) in (a) momentum space and (b) position space. A 405 nm pump beam generates degenerate, orthogonally polarized photon pairs via type-II SPDC in a ppKTP crystal. Interference of the generated photon is enabled by induced coherence. A knife edge is placed in the idler arm to perform edge measurements in either the far-field (a) or near-field (b) of the crystal. (a) Momentum-space configuration: All lenses used have 200 mm focal lengths. The pump is focused onto the crystal, generating SPDC photon pairs. DM2 separates the forward-propagating pump from the signal and idler photons. Lens L2 projects the crystal’s far-field onto mirror M1 and simultaneously focuses the back-propagating pump (reflected from M1) into the crystal. After the PBS, lenses L3 (signal arm) and L4 (idler arm) project the far-field onto mirrors M2 and M3, respectively. The knife edge is placed just before M3 in the idler arm, in the far-field plane of the crystal. The back-propagating signal and idler are separated from the pump by DM1, and lens L1 images the crystal’s far-field onto the EMCCD camera. A polarizer selects the signal photon polarization before detection. (b) Position-space configuration: All lenses have 100 mm focal lengths. The pump is focused onto the crystal, where SPDC photon pairs are generated. DM2 separates the forward-propagating pump from the down-converted photons. Lenses L3 and L4 image the crystal’s near-field onto M1 and simultaneously focus the back-propagating pump (from M1) into the crystal. Additional 4f imaging systems (L4–L5 for the signal arm and L4–L6 for the idler arm) project the near-field onto mirrors M2 and M3. The signal and idler photons are separated by a PBS placed after L4. The knife edge is positioned just before M3 in the near-field plane of the crystal. The back-propagating signal and idler are separated from the pump at DM1, and a 2f–2f imaging system using L1 maps the crystal’s near-field onto the EMCCD camera. A polarizer selects the detected signal photon.
• Figure 3: Experimental results for momentum space. (a) Represents the reconstructed visibility map of the knife edge -- an absorptive sharp object. (b) Illustration of the imaging process with the EMCCD camera capturing the detected beam intensity as the interferometric phase $\phi$ is scanned. The intensity varies with the phase change. By examining the recorded interference pattern on a pixel-by-pixel, independent information about each point on the object is derived. A sinusoidal function is fitted to the interference pattern of each pixel at various phases. This fitting process is demonstrated for a few pixels. The visibility for each pixel is determined from the obtained fit.
• Figure 4: Quantitative measurement of the edge-spread function in momentum and position space. (a) Cross-section of the reconstructed visibility image obtained from a sharp knife-edge placed in the momentum plane. The visibility profile is fitted with an error function, and the characteristic width $D_k$ is extracted from the 76%–24% rise interval. The measured value is $D_k = (352 \pm 17)\,\mu\mathrm{m}$, in good agreement with the theoretical prediction of $337\,\mu\mathrm{m}$. (b) Corresponding measurement in the position plane. From the error-function fit, we obtain $D_\rho = (28 \pm 3)\,\mu\mathrm{m}$, compared with the theoretical value of $21\,\mu\mathrm{m}$. The reduced visibilities in both measurements reflect the significant losses present in the idler arm; nonetheless, the extracted widths $D_k$ and $D_\rho$ remain unaffected by these losses, demonstrating the robustness of the method.