Testing Single Photon Entanglement using Self-Referential Measurements

TL;DR

This paper demonstrates a Bell-inequality violation using two identical copies of a single-photon entangled state, implementing self-referential measurements that exploit one copy as a phase reference for the other and avoiding homodyne detection. By performing joint measurements on pairs of modes ($A_1,A_2$) and ($B_1,B_2$) after binary phase choices, the experiment yields CHSH violations with and without post-selection, achieving $|\mathcal{B}|=2.23\pm0.07$ (no post-selection) and $|\mathcal{\tilde{B}}|=2.71\pm0.09$ (with post-selection) in line with theoretical predictions for the given indistinguishability $\mathcal{V}\approx 0.95$. The results establish a new, more accessible route to single-photon nonlocality that does not rely on shared local oscillators and hold potential for general mode-entangled states across photonic and massive-particle platforms. The approach highlights the role of two-copy, self-referential measurements as a practical tool for exploring fundamental quantum correlations and their applications in quantum information processing.

Abstract

Entanglement does not always require one particle per party. It was predicted some thirty years ago that a single photon traversing a beam splitter could violate a Bell inequality. Although initially debated, single-photon nonlocality was eventually demonstrated via homodyne measurements. Here, we present an alternate realisation that avoids the complexity of homodyne measurements and potential loopholes in their implementation. We violate a Bell inequality by performing joint measurements on two copies of the same single-photon entangled state, where one photon acts as a phase reference for the other, making it self-referential. We observe CHSH parameters of $2.71\pm 0.09$ and $2.23\pm 0.07$, depending on the joint measurements implemented. This offers a new perspective on single-photon nonlocality and a more accessible experimental route, potentially applicable to general mode-entangled states in diverse platforms.

Figures (4)

• Figure 1: Self-referential measurements. Our scheme requires two identical copies of a single-photon entangled state. Each copy is produced by sending a single photon to a beam splitter. The two modes of each single-photon entangled state are then shared between Alice and Bob, which they can use to violate a CHSH inequality by performing joint measurements on them. Prior to these joint measurements, there is no entanglement between photon 1 and photon 2. In this scheme, photon 2 can be seen to act as a phase reference for photon 1, and vice versa, making this a self-referential measurement.
• Figure 2: Experimental Setup. A source (orange field) generates spectrally identical photons $\gamma_1$ and $\gamma_2$, which are made indistinguishable in polarization by passing through a half-wave plate (HWP) and a linear polarizer (P) and are then individually placed in an entangled state using beam splitters (BSs). The two entangled states are then shared between Alice's lab (green field) and Bob's lab (magenta field) who each locally control a piezo-controlled phase shifter $(\phi_x, \phi_y)$, which implement their basis choices. Alice and Bob then perform joint measurements on their two photon modes, locally interfering them on a BS and using pseudo--number-resolving detectors ($D_{A_{1/2}}, D_{B_{1/2}}$) consisting of a fiber beam splitter and two detectors each, allowing them to violate a CHSH inequality.
• Figure 3: Bell Correlations. We measure the visibility of the Bell correlations, fixing Alice's phase at $\pi/8$ (green) and $-3\pi/8$ (magenta) -- corresponding to the two basis settings to maximize the CHSH parameter -- and scan Bob's phase. The Bell correlations oscillate with an amplitude of a.$0.902 \pm 0.011$ when not post-selecting and b.$0.914 \pm 0.012$ with post-selection, slightly lower than the measured HOM visibility. The black points indicate the location of the phase set points for Alice's and Bob's Bell basis choices $\phi_x, \phi_y \in \{\pi/8, -3\pi/8\}$ determined during calibration measurements similar to these data. c. The final Bell correlations and Bell parameters are determined in subsequent measurements setting the previously calibrated voltage values of the black points and measuring only these four Bell bases as opposed to the full scan. The estimated Bell correlations in each basis setting are compared to the theoretical predictions from Eq. (\ref{['eq:viz-corr-params']}) without post-selection and d. with post-selection.
• Figure 4: Alice and Bob compute the two-copy CHSH parameter by comparing their detections for their four basis choices. The maximally achievable parameter value depends linearly on the indistinguishability of the two photons. For perfect distinguishability, the Hong-Ou-Mandel (HOM) visibility is 1 and the maximal parameters are $1+\sqrt{2}$ without post-selection and $2\sqrt{2}$ with post-selection. We achieved an average visibility of 95% and a respective violation of $\mathcal{B} = 2.23 \pm 0.07$ and $\mathcal{\tilde{B}} = 2.71 \pm 0.09$, which is in good agreement with the theoretically achievable values of 2.293 and 2.687.