Imaging Non-local Bell type behavior | Progress of science

INTRODUCTION

Quantum entanglement makes an essential distinction between the classical world and the quantum world. The debate on the interpretation of this entanglement remained a central concern for much of the twentieth century. That an interpretation based on hidden variables can be ruled out based on an experimental observation is the very essence of Bell's inequality and follows the founding work of Freedman and Clauser (1) and Aspect et al. (2–4), many groups around the world have used non-local correlations between pairs of photons to show the violation of this type of inequality. Recent improvements to the design of the schema and the performance of the components have made it possible to simultaneously fill the various gaps present in the previous demonstrations (5–7).

The violation of a Bell inequality is a fundamental manifestation of a quantum system. Not only does it attest to the quantum fantasy of a system's behavior, but it also makes it possible to measure the performance of these systems when it is involved in certain quantum protocols. For example, some quantum protocols require non-local Bell-like behaviors, such as device-independent protocols (8–ten). Quantum Imaging is a quantum technology that is currently attempting to use the quantum behavior of light to achieve new types of imaging that can surpass the limits of conventional methods. As such, acquiring images of one of the most fundamental quantum effects is a demonstration that images can be exploited to access all the possibilities offered by the quantum world.

The violation of Bell's inequalities, opposing hidden variable interpretations of quantum mechanics, is usually based on the sequential measurement of correlation rates as a function of analyzer parameters (for example, relative angles of linear polarizers) acting on the two photons separated in space. One and the other. A violation of Bell 's inequality dictates that the correlation rate does not depend on the angle of one or the other polarizer, but of the combination of the two. The non-local nature of this correlation gives the name "remote phantasmagoric action". Polarization measurement is convenient, but entanglement tests have also been performed with other variables (11–14), and although the original Bell inequality has been applied to variables in a two-dimensional Hilbert space, a similar logic can be followed to design tests in space spaces. states with larger dimensions (15).

To understand our current work, it is important to consider two of the other large areas in which entanglement can be explored. The first large-scale domain is one that, instead of analyzing the polarization and, therefore, the angular momentum of photon spin, consists in measuring the orbital angular momentum (OAM) of photons. The photon OAM of provient comes from the helical phase structure of the beam described by exp (Iℓφ) (16, 17). Although the first experiments on this OAM focused on the observation of its mechanical manifestations (18), later work examined MAMA correlations between photons produced by parametric down-conversion showing entanglement (19) and, subsequently, a violation of a Bell-type inequality in two dimensions (20) and of higher dimension (21) OAM subspaces. The second large area concerns Einstein et al. (22), who expressed their concern about the completeness of quantum mechanics through the EPR paradox (Einstein-Podolsky-Rosen). This paradox concerns the correlations between the position and moment correlations that could be expected between the two entangled particles. For photon pairs produced by parametric downconversion, spatial correlations and moment anticorrelations can be observed in the image plane and the far field of the source, respectively. These correlations are the basis of quantum phantom imaging (23, 24), where one of the two down-converted beams is directed toward the object, with a single-pixel (non-spatial resolution) detector, collecting the interacting light, and the other beam directed to a detector at imaging. Data from one or the other detector alone does not give an image of the object, but a sum of correlations between non-imaging and imaging detectors gives . If the object and the detector with spatial resolution are in the image plans of the source, the image is vertical 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 vertical or inverted nature of the image stems from the position correlation and the moment anti-correlation, respectively, and can be considered as a manifestation of the image-based REP paradox, (25). In a different context, the demonstration of an EPR paradox in imaging was carried out in a series of experiments (26, 27, 28, 29) where cameras were used to record the position and momentum of entangled photon pairs. This eventually led to the demonstration of an EPR paradox within unique frames of a detector array (30). The question this work seeks to demonstrate is what type of imaging process might reveal Bell's inequality?

The key to the measurements in the polarization experiments is to recognize that the orientation of the linear polarizer analyzer is actually a measure of the phase difference between a superposition of right and left circular polarization states (ie say states of angular momentum of spin). . A right and left OAM overlay (= ± 1) has a phase step n on the beam diameter, the phase difference between the two OAM states defining the pitch orientation. Note that a circle contains all possible edge orientations and therefore a circle is equivalent to a full rotation of the polarizer in the case of polarization. This was exploited in a previous demonstration in which non-local phase step detection showed a contour-dependent contour enhancement, revealing a violation of Bell's inequality (31). However, this realization was not based on imagery because the data had been acquired sequentially by scanning an object in a spatially monomode optical configuration. The data from this set of measurements was finally recombined to be presented 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 been placed non-locally with respect to a circular phase object to give a ghost image, whose intensity characteristics reveal the anticipated violation of a bell. type inequality for OAM. This experiment does not require any scanning, relying on the use of a quantum state that can present correlations in the OAM space or in the Cartesian space direct. It shows that non – local Bell type behavior can be demonstrated in a full – field quantum imaging protocol. Because we do not close all the loopholes, our demonstration can not be interpreted as another absolute demonstration that the world behaves non-locally. However, these flaws are not fundamentally associated with the experimental paradigm presented here and could, in principle, be closed by more technically advanced phase image detectors and displays. Moreover, as we will see later, by making only a few physically reasonable assumptions about the source involved in the demonstration, our results can be interpreted as the first experimental demonstration that an imaging protocol can be used to reveal the type Bell type violation. behavior of a quantum system. Conversely, our results show that non-local Bell-like behavior can be exploited to achieve special types of imagery that could not be realized with a conventional conventional source.

RESULTS

Experimental editing and the principle of demonstration

Our experimental system, illustrated in Figure 1, consists of a β-barium borate crystal (BBO) pumped by a quasi-continuous 355 nm laser, thereby generating spatially entangled pairs of photons at 710 nm by the spontaneous parametric descent. conversion (SPDC). The two photons are separated on a beam splitter and propagate in two separate optical systems (arms). The first photon is reflected by a spatial light modulator (SLM) placed in an image plane of the crystal and displaying a phase object before being collected in a single mode fiber (SMF). It is then detected by a single photon avalanche diode (SPAD). The second photon, traveling through the other arm, is reflected by an SLM placed in a Fourier plane of the crystal (equivalent to the Fourier plane of the object) and displays a n-phase spatial pitch filter . The photon then propagates on an image-preserving delay line, approximately 20 m in length, before being detected by an intensified charge-coupled device (ICCD) camera. The ICCD camera is triggered conditionally to 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 camera trigger delays and ensures that the second photon is incident on the camera during the 4 ns gate time of the image intensifier.

We used such a trigger mechanism to implement a quantum illumination protocol and acquire images with less than one photon per pixel (32) and in the context of phase and amplitude imaging (33). We have also used a similar configuration to test the experimental limits of phantom imaging and ghost diffraction (34, 35). In the work currently reported, our scheme uses phase imaging to improve contours through spatial filtering. Here, the object (circular phase step) and the filter (straight-edge phase step) are placed non-locally in separate optical arms and are probed by two spatially separated but entangled photons. The image of the circle enhanced by the edges resulting from non-local interference between the object and the spatial filter probed by the two-photon wave function. However, the simple fact of obtaining an improved contour image under these circumstances is not in itself a proof of the non-local nature of the behavior of the two photons in that it can potentially be reproduced by conventional means such as in the context of ghost imaging (24). Nevertheless, images that can not be reproduced by conventional means can be produced by demonstrating the violation of a Bell inequality.

To understand that a Bell inequality can be violated by the implementation shown in FIG. 1, we can draw the conclusion that a stage π-phase includes both contributions ℓ = 1 and = -1, when they are expressed in OAM basis. In fact, a phase pitch π can be represented by the linear superposition of = -1 and = 1 and can therefore be represented on a Bloch-Poincaré sphere (36) describing a two-dimensional OAM base. In this context, the phase difference θ between the two modes ℓ = -1 and ℓ = 1 determines the orientation angle θ of the phase pitch π in the two-dimensional transverse plane. These steps can therefore be used as filters to perform measurements in this particular two-dimensional OAM space (20). Projected purely in such a space, the two-photon wave functions can be written in the following manner|ψ>=|–1>1|1>2+|1>1|–1>2(1)which is the result of keeping the total amount of MAO from the pump photons (= 0) to the signal and rest photons emitted by the SPDC process. Such a state will violate a Bell inequality of form (20)|S|≤2(2)withS=E(θ1,θ2)–E(θ1',θ2)+E(θ1,θ2')+E(θ1',θ2')(3)andE(θ1,θ2)=C(θ1,θ2)+C(θ1+π2,θ2+π2)–C(θ1+π2,θ2)–C(θ1,θ2+π2)C(θ1,θ2)+C(θ1+π2,θ2+π2)+C(θ1+π2,θ2)+C(θ1,θ2+π2)(4)or C(θ₁, θ₂) is the coincidence rate recorded when the first photon is detected after a phase with the θ orientation₁ and when the second photon is measured after a phase with the orientation θ₂. Inequality (Eq. 2) is a Bell inequality of Clauser-Horne-Shimony-Holt (CHSH) (37). As in a demonstration using the degree of freedom of polarization, the state (Eq.1) will exhibit a maximum violation of the inequality (Eq.2) when the settings will be chosen as follows: θ₁ = 22.5 °, θ1'=67.5°, θ₂ = 0 ° and θ2'=45°. In our implementation, all orientations θ₁ in arm 1 needed to perform the Bell test are obtained simply by using a two-dimensional circular phase step as the object displayed on the SLM 1. As can be seen in In Figure 1, it is necessary to have four different orientations for the spatial phase shift filter in the second arm (0 °, 45 °, 90 °, 135 °).

In our implementation, to perform an imaging of the Bell inequality, we used the reduced state (equation 1) in conjunction with the spatial correlations presented by the EPR state generated by SPDC to acquire a Spatially resolved image of Bell's behavior. We applied the phase filter in a Fourier plane of the crystal and placed the object in an image plane so that the filtering effect is applied to all the edges of the plane of the phase object, thus ensuring the simple taking of an advertised ghosting of the object will give us access to many parallel coincidence measurements on the ICCD camera, ie for the full range from 0 to 2π of θ₂ present in the object. Note that our intention here is not to target a test without loophole. The detector yields (~ 10% for the ICCD camera and ~ 50% for the SPAD) do not allow the closure of the detection fault; moreover, the technical triggering process of the camera used here means that the communication loophole is also not fulfilled in our implementation, since a conventional trigger signal is actually transmitted from one to the other. one detector to the other.

Finally, our demonstration does not provide randomization of the analyzer settings for both photons, which again leads to a loophole. In our experience, based on both imaging and projection in the OAM base, random adjustment of the phase filter orientation effectively provides a randomization of the second photon detection base. However, the use of a still image in the other branch 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 generation position of the photon pairs is also random and, more subtly, that this position is not bound, by an unknown process, in the state. OAM of light. While both of these assumptions are reasonable with respect to our entangled photon source, it should be noted that any claim of authentic non-local behavior depends on these assumptions. This caveat is the same for all demonstrations that are not flawless. For example, a detection flaw requires a correct sampling assumption (38). However, it should also be noted that, in our case, these warnings are imposed by technical limitations rather than by fundamental limitations. For example, the way in which the phase object is displayed can be changed for each shot before being reconstructed to allow a free choice of measurements made 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 random scan of the lateral position of the phase circle and then, after the measure, to "Scan" the associated component of the detected image. In the last part of Results, we report a successful implementation of these changes on the object displayed in arm 1. However, note that with existing technology, such a scan can not be done sufficiently quickly to overcome the gap in the locality.

Nevertheless, rather than aiming for a demonstration of non-localization without fundamental fault that has already been demonstrated (5–7), 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-like behavior of a quantum system. This makes it possible to perform the Bell test in the context of high dimensionality and with a highly parallel measurement acquisition method.

Violation of Bell's inequality in four images

In the first embodiment of this experiment, we acquired four distinct images corresponding to coincidence images of the ghost object filtered respectively by the four orientations, θ₂ = {0 °, 45 °, 90 °, 135 °} of the phase filter n. The images obtained directly by summing the thresholded frames acquired by the ICCD camera are illustrated in FIG. 2A. As previously discussed, these coincidence counting models will likely include the signature of a Bell-like behavior, and we can use these images to test for Bell's inequality (equation 2). To this end, it is possible to define a ring-shaped interest region (ROI) along the edge of the phase circle object in each of these images, as shown on the Figure 2 (B to E). We can then unfold these ROIs by defining classes of angles and radii and representing the images in polar coordinates. After integration along the different rays in the ROIs, we obtain the graphs presented in FIG. 2 (B to E), which respectively correspond to the four orientations, θ₂ = {0 °, 45 °, 90 °, 135 °} of the phase filter n. These graphs represent the coincidence counts as a function of the phase angle n θ₁ along the circle of phase. As can be seen, the experimental data extracted from the images closely follow the expected behavior of Malus in square sine, and one can test the Bell inequality (Eq.2) by selecting particular values ​​of θ₁ within these graphs. When selecting angles such as θ₁ = 22.5 °, θ1'=67.5°, θ₂ = 0 ° and θ2'=45°, Bell's inequality should be violated as much as possible.

By doing such a test of Bell, we findS=2.4626±0.0261(5)that is, results demonstrating a non-local Bell-like behavior of the two-photon state and separated from a classical behavior (S ≤ 2) by more than 17 DS.

Despite the fact that the classical limit of S > 2, the impeccable contrast obtained on the graphs presented in Fig. 2 (B to E) explains that the ultimate two-dimensional system 22 destination S is not saturated. This imperfect contrast comes from many factors. First, the imperfect spatial coherence of the two-photon interference due to the finite size of the SMF core (39) may result in lower contrast. Second, the technical noise of the camera associated with the presence of stray light can further reduce the contrast. Finally, the imperfect filtering of the phase circle by the phase filter can have a similar effect, even for a perfectly coherent imaging system using an ideal detector. This latter effect has been highlighted by the establishment of some simulations described in the supplementary documents.

Finally, it should be noted that the images presented in FIG. 2A are the results of interferometric filtering at a distance from the phase circle present in the arm 1 by the phase-shifted filter present in the arm 2. The type of imaging carried out here is more complex. than conventional ghost imaging schemes. We do not see any easy way to qualitatively reproduce our imaging results using only classical correlations, not to mention the quantitative violation of a Bell inequality that we report here, which requires entanglement.

Violation of Bell's inequality in a single image

In a second implementation of the experiment, we perform a demonstration of a violation of a Bell type inequality within a single accumulated image in order to demonstrate the ability of the l & # 39; s 39 quantum imaging to access highly dimensional parallel measurements. To observe the single-phase circle object filtered by the four different phase filters in a single image acquired by the camera, we add to the phase filters displayed on SLM 2 a different blazed network for each orientation θ₂ phase step of the phase filters. In this way, we deviate the beam in the arm 2 in a different way for each filter and thus obtain four simultaneous images of the phase circle in different parts of the photosensitive network of the camera. During the exposure time of each image acquired by the camera, we randomly select the phase mask displayed on the SLM 2 to switch between the four different phase filters θ₂ = {0 °, 45 °, 90 °, 135 °} randomly and with equal probability. The only image shown in FIG. 3A can then be accumulated, and by a similar processing of the images as above, defining the four ROIs shown in FIG. 3B, the curves of FIG. 3C expressing the coincidence counts as a function of FIG. θ₁ for the four different phase filter orientations θ₂.

Again, these data extracted from a single image can be used to test the Bell inequality (Eq.2). Using the following set of angles, θ₁ = 22.5 °, θ_1^' = 67.5 °, θ₂ = 0 ° and θ2'=45°, we findS=2,443±0,038(6)that is, demonstrate non-local behavior of Bell type in the single image. The results are, in the latter case, separated from the classical behavior by more than 11 DS.

Experimental realization with temporal displacement of the phase object

To be able to fill one of the gaps of our demonstration, we can introduce a displacement of the circle of variable phase as a function of the time displayed on the SLM 1 (in the arm 1 of the configuration) and apply a change of scenery corresponding to the detection of photons on the ICCD camera. So that the moving circle stays in the field of view of the arm 1, we have 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 the origin (0,0), the four possible positions in number of pixels are (10,10), (10, -10), (-10,10) and (-10, -10) . We then reproduce the same acquisition of individual images as that presented previously, except that for each of the images, a position of the phase circle is chosen and that we keep track of this position. A gross sum of the images thus acquired is presented in Fig. 4A. It can be seen that we still have four different parts in the image, each corresponding to the different orientations of the phase filter in the arm 2, but the expected filtered phase circles no longer appear because of the scanning of the circle. phase to different transverse positions. However, one can then use the information of the position of the phase circle to troubleshoot each of the images, then again to add all the images. The result is shown in Figure 4B, where we can see again the four distinctive filtered phase circles indicating a Bell Inequality test. This image can now be used in the same manner as described above to evaluate the Bell parameter. We findS=2,183±0.084(7)that is, demonstrate non-local behavior of Bell type in the single image. The results are, in the latter case, separated from the classical behavior by more than 2.17 DS.