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1.INTRODUCTIONThe central issue addressed by Bell’s work1 is about the correlation of results in spatially separated measurements. As clearly stated by Eberly,2 and most recently by Zeilinger,3 Bell inequalities do not derive from principles of quantum mechanics. They simply express an upper bound that a certain combination of correlation results should satisfy provided that our common sense ideas of realism and locality hold. The fact that there are systems violating Bell inequalities tell us that nature does not behave according to our everyday experience. Here, we report an extension of previous educational works4,5 that we believe is useful at the moment of illustrating the physical ideas behind Bell inequalities to the students. Our idea consists in measuring the S function for various experimental parameters in sharp contrast with previous experiments in which only a single value of the S function, the one where the Clauser-Horne-Shimony-Holt (CHSH) inequality6 is maximally violated, is measured. A particular characteristic of the experiment here reported is the fact that students themselves took the data working in pre-aligned setups in a two-hour laboratory session. This activity was part of a quantum optics course taken by both last year undergraduate and first year graduate students. 2.EXPERIMENTAL ACTIVITY WITH THE STUDENTSThe objective of the practice was to show the students that indeed, in nature, there are physical systems that violate an inequality of the type CHSH indicating that local realism has to be reconsidered. Photon’s polarization was chosen as the variable to measure the different outcomes, look at their correlations and find a value of S, which under a local realism theory is limited by In particular, paired photons in the Bell state For these reasons for the activity here reported, an instructor implemented a setup, as the one depicted in figure 1, which was later used by pairs of students, in two-hour laboratory sessions. As preparation for the session, students were asked to read an outreach explanation of the phenomena3 to introduce them to the background theory, and one paper about the experimental setup4 to introduce them to the technical details. Figure 1.Experimental setup for mapping the S function with entangled photons. Red lines depict the trajectory of the entangled photons. Left: schematic. Right: photograph of the actual setup; laser is outside the picture. ![]() For our implementation, a 405nm laser with output power of 30mW impinges a nonlinear crystal, a BBO, which has been cut and aligned to produce pairs of photons in the Bell state Students measured crossed correlations from coincident events between the polarization of photons for different angle configurations of the used HWPs. First they obtain the probability of measuring a coincident event for photon 1 under polarization X in a basis given by a HWP set at an angle α, and photon 2 under polarization Y in a basis given by its corresponding HWP set at an angle β; this probability is labeled as PXY(α, β). From these probabilities, a correlation function E(α, β) is defined as which is a quantity measured live during the session for a particular setting of HWPs, i.e. α and β. The goal of this practice is to find values of S such that equation (1) is violated. Function S is defined in terms of correlations in equation (2) given at two different settings of α and β, asLaboratory sessionQuantum mechanics predicts that for the A mapping of the S function is done by students, first by fixing a=0° and scanning b, and then by fixing a’=45° and scanning b as well. For our implementation we obtained a number of total coincidences of N=108cps (counts per second), students scanned the second HWP for 13 angles, and collected samples for 150s at each position. Including the time that took to rotate the waveplates, each scan lasted about 40 minutes, making possible to run the whole experiment without pressure under a 2 hour session. Students’ resultsAfter the laboratory session, students were asked to hand in a report which includes the theoretical calculation of the S function evaluated at the parameters used in the experimental session, i.e. S(0°,45°,b,b’), for the Each group reported their experimental results of the mapping of S by scanning variables b and b’. An example of obtained results is depicted in Figure 2. Main results obtained with the setup are listed on Table 1. Figure 2.Experimental results for the mapping of the S function by scanning angles of b. Marked points on grid represent experimental values for S obtained from the setup used in the practice. White region on the bottom of the surface shows values where S< -2, therefore violating Bell inequality, where several points lie. ![]() Table 1.Experimental results reported by students for the maximal measured value of S. Each of them violates equation (1).
The fact that one photon performs measurements for different HWP angles, preparing different projection states, allows observing the behavior of the S function, something that is not possible in studies that report just the value of S that violates optimally equation (1). This approach presents two advantages: First, it constitutes a method of state discrimination, and second, a method to measure if there is or not a violation of Bell inequality without the necessity of knowing the input state. 3.CONCLUSIONSWe report an experimental practice in which students are exposed to physical systems that motivates them to question and contrast their common sense ideas of locality and realism. Differently from other reports on this topic, the students reconstructed the CHSH-S function for different type of measurements getting an idea of its behavior and illustrating that a state can violate Bell inequality for some experimental settings and for others not. Additionally, the map of the CHSH-S function allows looking for violations of Bell inequality independently of the entangled state produced by the source. The reported experiments were performed by last year undergraduate and first year graduate students using pairs of photons generated via SPDC and considering the polarization state REFERENCESBell, J. S.,
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