Experimental setup and principle of the demonstration

Our experimental system, shown in Fig. 1, consists of a β-Barium Borate (BBO) crystal pumped by a quasi-continuous laser at 355 nm, thereby generating spatially entangled pairs of photons at 710 nm through the process of spontaneous parametric down-conversion (SPDC). The two photons are separated on a beam splitter and propagate into two distinct optical systems (arms). The first photon is reflected off a spatial light modulator (SLM) placed in an image plane of the crystal and displaying a phase object before being collected inside a single-mode fiber (SMF) and is subsequently detected by a single-photon avalanche diode (SPAD). The second photon, traveling through the other arm, is reflected off an SLM placed in a Fourier plane of the crystal (equivalent to the Fourier plane of the object) and displays a spatial π-phase step filter. The photon then propagates through a ~20-m-long, image preserving, delay line before being eventually detected by an intensified charge-coupled device (ICCD) camera. The ICCD camera is triggered conditionally on the detection of a photon by the SPAD placed in the first arm. This delay line ensures that the images obtained from the ICCD camera are coincidence images with respect to the SPAD detection. The presence of the delay line in the second arm compensates for the trigger delays of the camera and ensures that the second photon is incident on the camera during the 4-ns gate time of the image intensifier.

Fig. 1 Imaging setup to perform a Bell inequality test in images. A BBO crystal pumped by an ultraviolet laser is used as a source of entangled photon pairs. The two photons are separated on a beam splitter (BS). An intensified camera triggered by a SPAD is used to acquire ghost images of a phase object placed on the path of the first photon and nonlocally filtered by four different spatial filters that can be displayed on an SLM (SLM 2) placed in the other arm. By being triggered by the SPAD, the camera acquires coincidence images that can be used to perform a Bell test.

Such a triggering mechanism has been used to implement a quantum illumination protocol and acquire images with fewer than one photon per pixel (32), it was also used to infer the presence of entanglement in a polarisation entangled source after converting the polarisation state of one of the photons into spatial mode profile obtained by summing many individual photon events (33). The same triggering mechanism was used in the context of phase and amplitude imaging (34). We also used a similar setup to test the experimental limits of ghost imaging and ghost diffraction (35, 36). In the presently reported work, our scheme uses phase imaging to give edge enhancement through spatial filtering. Here, the object (a circular phase step) and the filter (a straight-edged phase step) are placed nonlocally within separated optical arms and are probed by two spatially separated but entangled photons. The resulting edge-enhanced image of the circle is a result of the nonlocal interference between the object and the spatial filter probed by the two-photon wave function. However, simply obtaining an edge-enhanced image in these circumstances is not in itself a proof of the nonlocal character of the two photons’ behavior in that it can potentially be reproduced by classical means as in the context of ghost imaging (24). One can nonetheless produce images that cannot be reproduced by classical means through the demonstration of the violation of a Bell inequality.

An understanding that a Bell inequality can be violated by the implementation presented on Fig. 1 can be drawn from the realization that a π-phase step has both ℓ = 1 and ℓ = −1 contributions when expressed in an OAM basis. A π-phase step can, in fact, be represented as the linear superposition of ℓ = −1 and ℓ = 1 and, thus, can be represented on a Bloch-Poincaré sphere (37) describing a two-dimensional OAM basis. In this context, the phase difference θ between the two modes ℓ = −1 and ℓ = 1 determines the orientation angle θ of the π-phase step in the two-dimensional transverse plane. One can therefore use these phase steps as filters to perform measurements in this particular two-dimensional OAM space (20). Projected purely into such a space, the two-photon wave functions can be written in the following way ∣ ψ 〉 = ∣ − 1 〉 1 ∣ 1 〉 2 + ∣ 1 〉 1 ∣ − 1 〉 2 (1)which is the result of the conservation of the total OAM from the pump photons (ℓ = 0) to the signal and idler photons emitted by the SPDC process. Such a state will violate a Bell inequality of the form (20) ∣ S ∣ ≤ 2 (2)with S = E ( θ 1 , θ 2 ) − E ( θ 1 ′ , θ 2 ) + E ( θ 1 , θ 2 ′ ) + E ( θ 1 ′ , θ 2 ′ ) (3)and E ( θ 1 , θ 2 ) = C ( θ 1 , θ 2 ) + C ( θ 1 + π 2 , θ 2 + π 2 ) − C ( θ 1 + π 2 , θ 2 ) − C ( θ 1 , θ 2 + π 2 ) C ( θ 1 , θ 2 ) + C ( θ 1 + π 2 , θ 2 + π 2 ) + C ( θ 1 + π 2 , θ 2 ) + C ( θ 1 , θ 2 + π 2 ) (4)where C(θ 1 , θ 2 ) is the recorded coincidence rate when the first photon is detected after a phase step with the orientation θ 1 and when the second photon is measured after a phase step with the orientation θ 2 . The inequality (Eq. 2) is a Clauser-Horne-Shimony-Holt (CHSH) Bell inequality (38). As in a demonstration using the polarization degree of freedom, the state (Eq. 1) will exhibit a maximal violation of the inequality (Eq. 2) when the settings are chosen in the following way: θ 1 = 22.5°, θ 1 ′ = 67.5 ° , θ 2 = 0°, and θ 2 ′ = 45 ° . In our implementation, all the orientations θ 1 in arm 1 necessary to perform the Bell test are obtained simply by using a two-dimensional circular phase step as the displayed object on SLM 1. As may be seen in Fig. 1, one needs to have four different orientations for the spatial phase step filter in the second arm (0°, 45°, 90°, 135°).

In our implementation, to perform imaging of the Bell inequality, we used the reduced state (Eq. 1) in conjugation with the spatial correlations exhibited by the EPR state generated through SPDC to acquire a spatially resolved image of the Bell behavior. We applied the phase filter in a Fourier plane of the crystal and placed the object in an image plane to ensure that the filtering effect will be applied to all the edges within the whole phase object plane, thus ensuring that simply taking a heralded ghost image of the object will give us access to many coincidence measurements in parallel across the ICCD camera, i.e., for the full 0 to 2π range of θ 2 present in the object. Note that our intention here is not to target a loophole-free test. The detector efficiencies (∼10% for the ICCD camera and ∼50% for the SPAD) do not allow the closing of the detection loophole; moreover, the technical triggering process of the camera used here means that neither is the communication loophole closed in our implementation because a classical trigger signal is actually conveyed from one detector to the other.

Last, our demonstration does not ensure the randomization of the analyzer settings for both photons, which leads again to a loophole. In our experiment, that is based both on imaging and on a projection in the OAM basis, the random setting of the phase filter orientation does ensure a randomization of the basis for the detection of the second photon. However, the use of a fixed image in the other arm means that it is the different spatial positions in the image that correspond to the different orientations of the phase step. For this second process to be random, we need to assume that the position of generation of the photon pairs is also random and, more subtly, that this position is not linked, by some unknown process, to the OAM state of the light. Although both of these assumptions are reasonable in relation to our source of entangled photons, it is noteworthy that any claim of genuine nonlocal behavior depends on these assumptions. This caveat is the same for all demonstrations that are not loophole free, for example, a detection loophole requires a fair sampling assumption (39). However, it is also to be noted that in our case, these caveats are imposed by technical limitations rather than by fundamental limitations. For example, the way the phase object is displayed can be varied for each shot before being reconstructed to lead to a free choice of measurements performed on each side. A possible approach to implement this and to break the link between the lateral position of the photon and the corresponding angle of the edge of the phase circle is to apply a randomized scan of the lateral position of the phase circle and then, after measurement, to “de-scan” the associated component of the detected image. In the last part of Results, we report a successful implementation of these changes to the object displayed in arm 1. However, note that with the existing technology, such a scan cannot be made sufficiently fast to overcome the locality loophole.

Nevertheless, rather than targeting a fundamental loophole-free demonstration of nonlocality that has already been demonstrated (5–7), here we aim to demonstrate that it is possible to use a full-field imaging system and quantum imaging tools and techniques to reveal the Bell-type–violating behavior of a quantum system. This allows the Bell test to be performed in the context of high dimensionality and with a highly parallel measurement acquisition method.