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Illustration of a nonlocal behavior of the type Bell | Advances in science



INTRODUCTION

The quantum entanglement makes a significant difference between the classical and the quantum world. The debate over the interpretation of this entanglement remained central for much of the 20th century. That an interpretation based on hidden variables can be excluded on the basis of experimental observation is the essence of Bell's inequality and follows the groundbreaking work of Freedman and Clauser ( 1 ) and Aspect et al. ( 2 4 ) many groups worldwide have used nonlocal correlations between photon pairs to show the violation of this kind of inequality. In particular, recent improvements to the schema design and performance of the components have made it possible to simultaneously close the various gaps that were present in previous demonstrations ( 5 ̵

1; 7 ).

The violation of a Bell's inequality is a fundamental manifestation of a quantum system. It not only confirms the quantum laxity of the behavior of a system, but also evaluates the performance of these systems when they are involved in certain quantum protocols. For example, certain quantum protocols require the performance of Bell-type non-local behaviors, such as device-independent protocols ( 8 10 ). One quantum technology that is currently of interest is quantum imaging, which attempts to exploit the quantum behavior of light in order to perform new types of imaging that can exceed the limits of classical methods. Capturing images of one of the most basic quantum effects is therefore evidence that images can be used to access the full range of possibilities allowed in the quantum world.

The Violation of Bell's Inequalities Against Hidden Variable Interpretations of Quantum Mechanics has usually relied on the sequential measurement of correlation rates as a function of analyzer settings (eg, the relative angles of the linear polarizers) that rely on the act on two separate photons in space. A violation of Bell's inequality causes the correlation rate to depend not only on the angle of a polarizer, but on the combination of the two. Due to the non-locality of this correlation, the term "spooky distance effect" is used. Measurement of polarization is convenient, but interlacing tests were also performed on other variables ( 11 14 [19459006)]) and although the original Bell inequality was applied to variables in a two-dimensional Hilbert space , a similar logic can be used for design tests in higher-dimensional state spaces ( 15 ).

To understand our present work, it is important to consider two of the other high-dimensional realms in which entanglements can be explored. The first high-dimensional domain is that instead of analyzing the polarization and thus the spin angular momentum of the photons, alternatively the orbital angular momentum (OAM) of the photons is measured. The OAM of ℓℏ per photon results from the helical phase structure of the beam described as exp ( i ℓφ) ( 16 17 .) Although early experiments with this OAM are based on observation of his mechanical manifestations ( 18 ), later works examined the correlations of the OAM between photons generated by parametric downconversion and entanglement ( 19 ) and subsequently a violation of a Bell's inequality in two-dimensional ( 20 ) and higher-dimensional ( 21 ) OAM subspaces.) The second high-dimensional domain concerns Einstein et al. ( 22 ), who express their concern about the Completeness of quantum mechanics was expressed by the EPR paradox (Einstein-Podolsky-Rosen) This paradox concerns the correlations between position and imp uls correlations that could occur between the two entangled particles. For photon pairs generated by parametric downconversion, both spatial correlations and impulse anticorrelations can be observed in the image plane and in the far field of the source, respectively. These correlations form the basis for quantum ghost imaging ( 23 24 ), where one of the two down-converted beams is directed to the object with a single pixel (not spatially resolving). Detector, which collects the interacting light, and the other beam directed to an imaging detector. The data from neither detector yields an image of the object, but a summation of the correlations between the non-image detector and the image detector. When the object and the spatial resolution detector are in the image planes of the source, the image is upright with respect to the object; If the object and the detector are in the far field of the source, the image is inverted with respect to the object. The upright or inverted nature of the image results from the position correlation or the impulse anticorrelation and can be regarded as an image-based manifestation of the EPR paradox ( 25 ). In another context, the demonstration of an EPR paradox in imaging has been performed in a series of experiments ( 26 27 28 29 ) in which cameras were used to perform the Position and impulse of entangled photon pairs. This eventually led to the demonstration of an EPR paradox within individual frames of a detector array ( 30 ). The question to be presented in this work is what kind of imaging could reveal a Bell's inequality.

The key to measurements in the polarization experiments is to recognize that the orientation of the linear polarizer analyzer is actually a measurement of the phase difference between a superposition of the right and left circular polarization states (ie the angular momenta states). A superposition of right and left OAMs (ℓ = ± 1) has a π-phase step across the diameter of the beam, with the phase difference between the two OAM states determining the orientation of the step. Note that a circle contains all possible edge orientations and therefore as an object a circle corresponds to a full rotation of the polarizer in the polarization case. This was exploited in an earlier demonstration in which the non-local detection of phase steps showed an edge enhancement depending on the edge orientation, indicating a violation of Bell's inequality ( 31 ). However, this finding was not based on imaging because the data was sequentially acquired by scanning an object in a single-mode spatial optical setup. The data from this set of measurements was finally recombined to be displayed in the form of an image. In contrast, in this work, we use a full field imaging configuration in which a phase edge filter has not been placed locally with respect to a circular phase object to obtain a ghost image whose intensity characteristics reveal the expected violation of a bell inequality for OAM. No sampling is required for this experiment, since a quantum state must be used which can exhibit correlations in OAM space or in direct Cartesian space. It shows that bell-type non-local behavior can be demonstrated within a full field quantum mapping protocol. Since we do not fill all the gaps, our demonstration can not be interpreted as another absolute demonstration that the world does not behave locally. However, these gaps are not fundamentally associated with the experimental paradigm presented here and could, in principle, be concluded with more technically advanced detectors and phase-image displays. In addition, as will be discussed below, our results may be interpreted as the first experimental demonstration that an imaging protocol can be used to uncover the Bell-type violation by making only some physically reasonable assumptions about the source involved in the demonstration become behavior of a quantum system. Conversely, our results indicate that Bell-type non-local behavior can be exploited to perform special types of imaging that could not be performed with a conventional classical source.

RESULTS

Experimental Design and Principle of Demonstration

Our experimental system shown in Figure 1 consists of a β-barium borate (BBO) crystal pumped at 355 nm using a quasicontinuous laser to give spatially interlaced Photon pairs at 710 nm were generated by the process of spontaneous parametric downconversion (SPDC). The two photons are separated on a beam splitter and spread out into two different optical systems (arms). The first photon is reflected by a spatial light modulator (SLM) located in an image plane of the crystal and displaying a phase object before being collected in a single-mode fiber (SMF) followed by a single-photon avalanche diode (SPAD). The second photon moving through the other arm is reflected by an SLM located in a Fourier plane of the crystal (corresponding to the object's Fourier plane) and displays a spatial π-phase step filter. The photon then propagates through approximately 20 meters of image-sparing delay line before being finally detected by an ICCD (Intensified Charge Coupled Device) camera. The ICCD camera is triggered on the condition that the SPAD in the first arm detects a photon. This delay line ensures that the images obtained from the ICCD camera are match images with respect to the SPAD recognition. The presence of the delay line in the second arm compensates for the camera's shutter lag and ensures that the second photon falls onto the camera during the 4 ns gate of the image intensifier.

Fig. 1 Image construction for carrying out a Bell inequality test in pictures.

A BBO crystal pumped with an ultraviolet laser is used as a source of intertwined photon pairs. The two photons are separated on a beam splitter (BS). An amplified camera triggered by a SPAD is used to capture ghosts of a phase object placed on the path of the first photon and through four different spatial filters that can be displayed on one SLM (SLM 2) in the other arm , is not filtered locally. Triggered by the SPAD, the camera captures random images that can be used to conduct a Bell test.

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Fig. 1 Image construction for performing a Bell inequality test in images.

As source for entangled photon pairs Using a BBO crystal pumped with an ultraviolet laser, the two photons are separated on a beam splitter (BS) .A amplified camera triggered by a SPAD is used to capture ghosts of a phase object traveling along the path of the first Photons are not spatially filtered by four different spatial filters that can be displayed on one SLM (SLM 2) in the other arm, and by triggering with the SPAD, the camera captures random images that can be used to conduct a Bell test.

Using such a triggering mechanism, we have implemented a quantum illumination protocol and captured images with less than one photon per pixel ( 32 ) and in the context of phase unis d Amplitude images ng ( 33 ). We also used a similar setup to test the experimental limits of ghost imaging and ghost diffraction ( 34 35 ). In the present work, our scheme uses phase mapping to improve the edges by spatial filtering. Here, the object (a circular phase step) and the filter (a rectilinear phase step) are not located locally in separate optical arms and examined by two spatially separated but entangled photons. The resulting edge-enhanced image of the circle is a result of the non-local interference between the object and the spatial filter being examined by the two-photon wave function. However, simply obtaining an edge enhanced image under these circumstances is not evidence of the nonlocal nature of the behavior of the two photons, as it may possibly be reproduced by classical means such as in the context of ghost imaging ( 24 ). By proving the violation of a Bell inequality, one can still produce images that are not reproducible by classical means.

From the realization that the implementation shown in Fig. 1 can violate Bell's inequality, it is possible to derive the understanding. A π-phase step has both ℓ = 1 and ℓ = -1 contributions if it is in an OAM Basis is expressed. Indeed, a π-phase step can be represented as a linear superposition of ℓ = -1 and ℓ = 1, and thus on a Bloch-Poincaré sphere ( 36 ), which describes a two-dimensional OAM basis. The phase difference θ between the two modes ℓ = -1 and ℓ = 1 determines the orientation angle θ of the π-phase stage in the two-dimensional transverse plane. One can therefore use these phase steps as a filter to perform measurements in this particular two-dimensional OAM space ( 20 ). Projected purely into such a space, the two-photon wavefunctions can be written as follows

] ] = | 1 [19659026]> 1 [19659032]> 1 > 2 + > 1 > 1 | 1 > 2

(1) . This is the result of preserving the entire OAM from the pump photons (ℓ = 0) to those from the SPDC process. Such a state violates a Bell's inequality of the form ( 20 )

| S | 2

(2) with [19659055] S = E ( θ 1 θ 2 ) E [19659021] ( θ 1 ¢ θ 2 ) + [E ( θ 1 θ 2 [) + [E ( θ 1 [[19659065] θ 2 # )

(3) 19659046] and

E ( θ 1 θ ] 2 ) = C [19659050] & thgr; 1 [19659059]? 2 ) + ( θ 1 + π 2 [19659059] 2 + 2 ) C ( θ 1 + π 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 + [19659113] π 2 θ 2 [1 9659129] + ] C ( θ 1 θ 2 + π 2

(4)

where C 1 θ 2 ) is the recorded coincidence rate when the first photon is detected thereafter a phase step with the orientation θ 1 and when the second photon after a phase step with the orientation θ 2 is measured. The inequality ( Eq. 2) is a Clauser-Horne-Shimony-Holt (CHSH) Bell inequality ( 37 ). As in a demonstration with the degree of polarization freedom, the condition (Equation 1) shows a maximum violation of the inequality (Equation 2) when the settings are chosen as follows: θ 1 = 22.5 °,

& thgr; ; 1 [ = 67.5 °

& thgr; 2 = 0 ° and

& thgr; 2 [ = 45 °

. In our implementation, all orientations .theta. 1 in arm 1, which are required to perform the Bell test, is obtained simply using a two-dimensional circular phase step as a displayed object on SLM 1. Fig. In Fig. 1 the spatial phase step filter in the second arm must have four different orientations (0 °, 45 °, 90 °, 135 °).

In our implementation, we have used the reduced state (equation 1) to map the Bell inequality, in conjunction with the spatial correlations exhibited by the SPR generated EPR state, to obtain a spatially resolved image of the Bell behavior. We have applied the phase filter in a Fourier plane of the crystal and placed the object in an image plane to ensure that the filter effect is applied to all edges in the entire phase object plane, ensuring that simply an announced ghost image is taken of the object allows parallel access to many match measurements with the ICCD camera, ie for the entire 0 to 2π range of θ 2 present in the object. Note that we do not intend to perform a gapless test here. Detector efficiencies (~ 10% for the ICCD camera and ~ 50% for the SPAD) do not allow the recognition gap to close; In addition, the technical triggering process of the camera used here means that in our implementation, no communication gap is closed because a classic triggering signal is actually transmitted from one detector to another.

Finally, our demonstration does not guarantee the randomization of the analyzer's settings for both photons, which in turn leads to a gap. In our experiment, which is based both on the mapping and on a projection in the OAM basis, the random adjustment of the phase filter orientation ensures a random distribution 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 must assume that the position of the generation of the photon pairs is also random and, more subtly, that this position is not linked by an unknown process to the OAM state of the light. Although both assumptions regarding our source of entangled photons are reasonable, it is noteworthy that any assertion of true nonlocal behavior depends on these assumptions. This restriction applies to all incomplete demonstrations. For example, a recognition gap requires a fair sampling ( 38 ). However, it should also be noted that these limitations in our case are due to technical rather than fundamental limitations. For example, the way the phase object is displayed may be varied for each shot before it is reconstructed to allow a free choice of measurements made on each side. One possible approach to doing this and solving the connection between the lateral position of the photon and the corresponding angle of the edge of the phase circle is to perform a randomized scan of the lateral position of the phase circle and then to scan it after the measurement recognized image. In the last part of the results we report a successful implementation of these changes on the object displayed in arm 1. Note, however, that with existing technology, such a scan can not be performed fast enough to close the locality gap.

Nevertheless, instead of aiming at a basic uninterrupted demonstration of non-locality that has already been demonstrated ( 5 7 ), let us demonstrate here that it is possible to use a full field Imaging system and quantum imaging tools and techniques for detecting the Bell-type infringing behavior of a quantum system. This allows the Bell test to be performed in the context of high dimensionality and with a highly parallel data acquisition method.

Violation of the Bell Inequality in Four Images

In the first run of this experiment, four separate images were taken correspondingly. Match images of the ghost object, each filtered through the four orientations, & thetas; 2 = {0 °, 45 °, 90 °, 135 °} of the & pgr; -Phasenfilters. The images obtained directly by summing up the thresholds detected by the ICCD camera are shown in FIG. 2A. As already mentioned, these coincidence counting patterns probably contain the signature of the Bell behavior, and we can use these images to test the Bell inequality (Equation 2). For this purpose, one may define in each of these images along the edge of the phase circle object an annular region of interest (ROI) as shown in Fig. 2 (B to E). These ROIs can then be unfolded by defining angle and radius boxes and displaying the pictures in polar coordinates. After integration along the different radii within the ROIs, one obtains the graphs shown in FIG. 2 (B to E) corresponding respectively to the four orientations θ 2 = {0 °, 45 °, 90 °, 135 °} of the π-phase filter. These graphs represent the coincidence numbers as a function of the π phase angle θ 1 along the phase circle. As can be seen, the experimental data extracted from the images closely follows the expected sinusquadratic penalty-like behavior, and one can test the Bell inequality (Equation 2) by taking certain values ​​of θ 1 within selects these graphics. If the angles are chosen such that .theta. 1 = 22.5 °,

& thgr; 1 # = 67.5 °

θ 2 = 0 ° and

θ 2 [ = 45 ° ]

[19659046] it is expected that the Bell inequality will be violated as much as possible.

Fig. 2 Full-frame images that record the violation of Bell's inequality in four images.

( A ) The four coincidence counters corresponding to images of the phase circle taken with the four-phase filter with different orientations, θ 2 = {0 °, 45 ° , 90 °, 135 °}, required to perform the Bell test. Scale bar, 1 mm (in the plane of the object). ( B to E ) The coincidence counts graphs as a function of the orientation angle θ 1 of the phase step along the object. As shown, these results are obtained by unfolding the ROIs depicted as red rings and extracting them from the images shown in (A). The blue dots in the graphs are the match counts per angular range within the ROIs, and the red graphs correspond to the best fits of the experimental data through a cosine squared function. (B) to (E) correspond to phase filter orientations .theta. 2 of 0 °, 45 °, 90 ° and 135 °, respectively.

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Fig. 2 Full-screen images that record the violation of a Bell inequality in four images.

( A ) The four coincidences count Images corresponding to images of the phase circle taken with the four phase filters of different orientations, θ 2 = {0 °, 45 °, 90 °, 135 °} required to carry out the bell test scale bar, 1 mm (in the plane of the object). ( B to E ) The correspondence matters Graphs are plotted as a function of the orientation angle θ 1 of the phase step along the object As shown, these results are obtained by unfolding the ROIs represented as red rings and extracting them from the images shown in (A) in the diagrams are the coincidenceza hl s per angular range within the ROIs, and the red curves correspond to the best fits of the experimental data through a cosine squared function. (B) to (E) correspond to phase filter orientations .theta. 2 of 0 °, 45 °, 90 ° and 135 °, respectively.

When one passes to such a Bell test one obtains [19659241] S = 2.4626 ± 0.0261

(5) ie results which are nonlocal Behavior of the Bell type of the two-photon state show and are separated from a classical behavior ( S ≤ 2) by more than 17 SD.

Despite exceeding the classical limit of S > 2, the not perfect contrast was obtained on the diagrams shown in . 2 (B to E) states that the final two-dimensional

2 2

is not intended for S saturated , This incomplete contrast results from several factors. First, the imperfect spatial coherence of two-photon interference due to the finite size of the SMF core ( 39 ) may lead to lower contrast. Second, the camera noise together with the presence of parasitic light can further reduce the contrast. Finally, the incomplete filtering of the phase circuit by the phase filter can lead to a similar effect, even for a perfectly coherent imaging scheme using an ideal detector. The latter effect has been substantiated by the creation of some simulations described in Supplementary Materials.

Finally, it is noteworthy that the images shown in Fig. 2A are the results of the remote interferometric filtering of the phase circuit present in arm 1. Here, the type of imaging performed here is more complex than in conventional ghost imaging schemes. We see no easy way to qualitatively reproduce our imaging results with classical correlations, let alone the quantitative violation of a Bell inequality that requires entanglement.

Bell inequality violation in a single image

In One Second In performing the experiment, we demonstrate the violation of a Bell's inequality in a single accumulated image to demonstrate the ability of quantum imaging to access high-dimensional parallel measurements. To observe the single-phase circle object filtered by the four different phase filters in a single frame captured by the camera, we add another blazed grating for each orientation to the phase filters shown in the SLM 2 phase filter. Auf diese Weise lenken wir den Strahl in Arm 2 für jedes Filter auf unterschiedliche Weise um und erfassen daher vier gleichzeitige Bilder des Phasenkreises in verschiedenen Teilen des lichtempfindlichen Arrays der Kamera. Während der Belichtungszeit jedes von der Kamera erfassten Bildes wählen wir zufällig die auf dem SLM 2 angezeigte Phasenmaske aus, um zwischen den vier verschiedenen Phasenfiltern umzuschalten. Θ 2 = {0 °, 45 °, 90 °, Zufällig und mit gleicher Wahrscheinlichkeit. Man kann dann das in Fig. 3A gezeigte Einzelbild akkumulieren und durch eine ähnliche Behandlung der Bilder wie zuvor, wobei die in Fig. 3B gezeigten vier ROIs definiert werden, erhält man die Kurven in Fig. 3C, die die Übereinstimmungszahlen als Funktion von ausdrücken θ 1 für die vier verschiedenen Phasenfilterorientierungen θ 2 .

Abb. 3 Vollbild-Einzelbild, das die Verletzung einer Bellschen Ungleichung aufzeichnet.

( A ) Das durch unser Protokoll erfasste Koinzidenz-Einzelbild entspricht einem Bild derselben Phase Kreis, der mit den vier Phasenfiltern mit unterschiedlichen Ausrichtungen erfasst wurde, θ 2 = {0 °, 45 °, 90 °, 135 °}, der zur Durchführung des Bell-Tests erforderlich ist. Maßstabsbalken, 1 mm (in der Ebene des Objekts). ( B ) Die Korrespondenz zwischen den verwendeten Phasenfiltern und der jeweiligen Beobachtung des im Einzelbild erfassten Objekts wird hervorgehoben. Die vier zur Behandlung des Einzelbilds verwendeten ROIs sind in (B) ebenfalls hervorgehoben. ( C ) Die Koinzidenzzahlkurven als Funktion des Orientierungswinkels θ 1 des Phasenschritts entlang des Objekts für die vier verschiedenen Orientierungen der Phasenfilter werden dargestellt. Diese Grafiken werden allein durch Extrahieren der Übereinstimmungszahlen in dem in (A) dargestellten Einzelbild erhalten.

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3 Vollbild-Einzelbild, das die Verletzung einer Bellschen Ungleichung aufzeichnet.

( A ) Es wird das durch unser Protokoll erfasste Koinzidenz-Einzelbild dargestellt, das einem Bild desselben Phasenkreises entspricht erfasst mit den vier Phasenfiltern mit unterschiedlichen Ausrichtungen, θ 2 = {0 °, 45 °, 90 °, 135 °}, erforderlich zur Durchführung des Bell – Tests. Maßstab 1 mm (in der Ebene von das Objekt). ( B ) Die Korrespondenz zwischen den verwendeten Phasenfiltern und der jeweiligen Beobachtung des im Einzelbild erfassten Objekts wird hervorgehoben. Die vier zur Behandlung des Einzelbilds verwendeten ROIs werden auch in (B) hervorgehoben ). ( C ) Der Zufall zählt Graphen als Funktion des Orientierungswinkels θ 1 des ph Ein Schritt entlang des Objekts für die vier verschiedene n Ausrichtungen der Phasenfilter wird dargestellt. Diese Graphen werden allein durch Extrahieren der Koinzidenzzahlen in dem in (A) dargestellten Einzelbild erhalten.

Wiederum kann man diese aus dem Einzelbild extrahierten Daten verwenden, um einen Test der Bellschen Ungleichung durchzuführen (Gl. 2). Unter Verwendung des folgenden Winkelsatzes ist & thgr; 1 = 22,5 °, & thgr; 1 ' = 67,5 °, & thgr; 2 = 0 ° und [19659265] θ 2 ' = 45 °

ein Fund

S = [19659025]2.443±0.038

(6)that is, demonstrating a Bell-type nonlocal behavior in the single image. The results are, in that latter case, separated from classical behavior by more than 11 SDs.

Experimental realization with time-varying displacement of the phase object

To be able to close one of the existing loopholes in our demonstration, we can introduce a time-varying displacement of the phase circle displayed on SLM 1 (in arm 1 of the setup) and apply a corresponding de-scan to the photon detection on the ICCD camera. To keep the moving circle within the field of view in arm 1, we slightly reduced its size to a radius of 21 pixels. The circle is moved between four different possible positions around the center of the beam. Taking the center of the beam as origin (0,0), the four possible positions in numbers of pixels are (10,10), (10,−10), (−10,10), and (−10,−10). We then reproduce the same acquisition of single images as presented previously, with the difference now being that, for each of the images, a position of the phase circle is chosen, and we keep track of this position. A raw sum of the images thus acquired is presented in Fig. 4A. One can observe that we have still four different parts in the image, each corresponding to the different orientations of the phase filter in arm 2, but the expected filtered phase circles do not appear anymore because of the scanning of the phase circle to different transverse positions. However, one can then use the information of the position of the phase circle to de-scan each of the images and then again summing all of the images together. The result is shown in Fig. 4B, where one can see once again the four distinctive filtered phase circles indicative of a test of a Bell inequality. One can now use this image in the same way as described previously to perform an evaluation of the Bell parameter. We find

S=2.183±0.084

(7)that is, demonstrating a Bell-type nonlocal behavior in the single image. The results, are in that latter case, separated from classical behavior by more than 2.17 SDs.

Fig. 4 Full-frame single image recording the violation of a Bell inequality and implementing the scanning of the phase circle.

(A) The raw sum of the coincidence counting single image acquired through our protocol is presented, which corresponds to an image of the same phase circle acquired with the four phase filters with different orientations, θ2 = {0° , 45° , 90° , 135° }, necessary to perform the Bell test. (B) The image obtained by de-scanning each of the images given the chosen position for the phase circle is presented. We can use this latter image to perform an evaluation of the Bell parameter S and to demonstrate the nonlocal behavior.

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Fig. 4 Full-frame single image recording the violation of a Bell inequality and implementing the scanning of the phase circle.

(A) The raw sum of the coincidence counting single image acquired through our protocol is presented, which corresponds to an image of the same phase circle acquired with the four phase filters with different orientations, θ2 = {0° , 45° , 90° , 135° }, necessary to perform the Bell test. (B) The image obtained by de-scanning each of the images given the chosen position for the phase circle is presented. We can use this latter image to perform an evaluation of the Bell parameter S and to demonstrate the nonlocal behavior.

MATERIALS AND METHODS[19659296]The four images shown in Fig. 3A and the single image shown in Fig. 2A were each obtained by acquiring 40,000 frames each of 1 s of exposure, during which time the camera intensifier was triggered for every heralding detection by the SPAD. The ICCD sensor was air cooled to −30°C. The images were thresholded to generate binary images that correspond to the detection of single photons. We calculated the threshold over which a pixel was considered to correspond to a photo-detection and the noise probability per pixel by acquiring 5000 frames with the camera optical input blocked. The dark count probability per pixel and per frame arising from the camera readout noise was then calculated to be around 5 × 10−5.

The images obtained correspond to photon correlation images, the intensity in the images corresponds to the number of coincidence counts because the camera is triggered by the detection of the first photon by the SPAD, and the images are then analyzed to test the Bell inequality. First, we located the center of each image circle, and we defined ring-like ROIs to follow the edges of each object. These rings are 17 pixels in width with a mean radius of 26 pixels. The coincidence counting images within the ROIs were then converted into polar coordinates. We used 48 angular bins from 0 to 2π, and we integrated over the 17-pixel width of the ROIs to obtain the coincidence as a function of the angle θ1 corresponding to the local orientation of the π-phase step at a particular position on the camera. From these data points, one can read the coincidence rates corresponding to the angles of interest to perform the Bell test.

Uncertainties on the mean value of S were obtained as SEs by splitting the set of 40,000 frames into 20 parts of 2000 frames and evaluating for each of the 20 sets a value for S. With these 20 values of Swe then computed the means and the SEM. Note that a detailed schematic of the experimental setup is available in the Supplementary Materials.

Acknowledgments: Funding: P.-A.M. acknowledges the support from the European Union Horizon 2020 Research and Innovation Program under the Marie Sklodowska-Curie Action (Individual MSCA Fellowship no. 706410) of the Leverhulme Trust through the Research Project Grant ECF-2018-634 and of the Lord Kelvin/Adam Smith Leadership Fellowship Scheme. M.J.P. acknowledges the financial support from the EPSRC QuantIC (EP/M01326X/1) and the ERC TWISTS (340507). R.S.A., E.T., T.G., and P.A.M. acknowledge the financial support from the UK EPSRC (EP/L016753/1 and EP/N509668/1). T.G. acknowledges support from the Professor Jim Gatheral Quantum Technology Studentship. Author contributions: M.J.P. initiated the project. P.-A.M. and M.J.P. conceptualized the demonstration and interpreted the results. P.-A.M., R.S.A., and M.J.P. designed the experiment. P.-A.M. conducted the experiment with help from E.T., T.G., P.A.M., and R.S.A. and developed the analysis tools. All authors contributed to the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper will be made available online in the University of Glasgow repository (http://researchdata.gla.ac.uk/).

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