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Izvestiya of Altai State University

Deterministic Interpretation of the Malus Law and Correlation in Experiments with Entangled Photons

Александр Иванович Гончаров

Published Mar 28, 2023• 0 citations
Publisher

Abstract

This paper examines a local deterministic model of an experiment involving correlated photons. The experimental setup involves a source that emits two linearly polarized photons in each act, two two-channel analyzers, and detectors included in the coincidence scheme. The orientation angle of the photon polarization plane is a random variable with a uniform distribution, and it is the same for simultaneously emitted photons. The model presented here is a generalization of the “naive example of hidden variable theory” described in A. Aspect's article ‘Bell's Theorem: The Naive View of an Experimentalist. In our model, the outcome of the photon interaction with the analyzer (hit in one of the two channels) is uniquely determined by the angle between the photon polarization plane and the analyzer axis and is described by a step function. The difference in our model is that the step function contains a large number of segments of different lengths. The location and lengths of the segments are set so that the probability of a photon hitting a specific channel of an ideal analyzer, determined by averaging the outcomes over small neighborhoods of angles, approaches the Malus law when the maximum length of a segment decreases. This ensures that the classical model is self-consistent. When ideal analyzers are used, the absolute value of the correlation coefficient of detector readings mainly does not exceed the values from Aspect's ‘naive example’. However, when analyzers with absorption are used, the calculated correlations for some orientations of the analyzers exceed the quantum mechanical values. Key words: linear polarization of photons, Malus law, entangled photons, correlations, hidden parameters.

DOI
10.14258/izvasu(2023)1-03

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This paper examines a local deterministic model of an experiment involving correlated photons. The experimental setup involves a source that emits two linearly polarized photons in each act, two two-channel analyzers, and detectors included in the coincidence scheme. The orientation angle of the photon polarization plane is a random variable with a uniform distribution, and it is the same for simultaneously emitted photons. The model presented here is a generalization of the “naive example of hidden variable theory” described in A. Aspect's article ‘Bell's Theorem: The Naive View of an Experimentalist. In our model, the outcome of the photon interaction with the analyzer (hit in one of the two channels) is uniquely determined by the angle between the photon polarization plane and the analyzer axis and is described by a step function. The difference in our model is that the step function contains a large number of segments of different lengths. The location and lengths of the segments are set so that the probability of a photon hitting a specific channel of an ideal analyzer, determined by averaging the outcomes over small neighborhoods of angles, approaches the Malus law when the maximum length of a segment decreases. This ensures that the classical model is self-consistent. When ideal analyzers are used, the absolute value of the correlation coefficient of detector readings mainly does not exceed the values from Aspect's ‘naive example’. However, when analyzers with absorption are used, the calculated correlations for some orientations of the analyzers exceed the quantum mechanical values. Key words: linear polarization of photons, Malus law, entangled photons, correlations, hidden parameters.
The paper considers a local deterministic phenomenological model of an experiment with correlated photons. A source emits two same-state photons simultaneously in each act. Then each photon enters its own two-channel analyzer. There are two parameters that characterize the photon state in this model. The first one is the polarization plane orientation angle. The second parameter is an “additional” one that takes values within the [0,l] range. When the photon interacts with the analyzer, it either gets into the “+” or “-” channel or is absorbed in the analyzer. Such an outcome is determined by the angle between the photon polarization plane and the axis of the analyzer along with the value of the “additional” parameter. The probability of a photon entering the "+" channel P{A= l} obeys the Malus law when considering a single-particle experiment with polarized photons and no photon absorption. And for a two-particle experiment, the dependence of the correlation coefficient of detectors readings on the angles between the analyzers’ axes follows the classic linear law. However, the correlation coefficient is described by a polyline that satisfactorily approximates the exact function when absorption is considered, and the probability P{A=l} for individual polarized photons is quite close to the Malus law.
Bell inequality experiments have demonstrated that quantum entanglement violates the predictions of local realism. However, all Bell experiments are based on joint measurements of two photons, and their analysis implicitly assumes that "the polarization state of a single photon is ideally linear." This paper proposes an experimental scheme to directly test this assumption: using type-II spontaneous parametric down-conversion (SPDC) to generate polarization-entangled photon pairs, employing fixed-angle measurement of the e-light channel as a coincidence gate to filter entangled photons in the o-light channel, then scanning the o-light analyzer angle with high precision to measure the coincidence count rate as a function of the analyzer angle, testing whether it strictly follows Malus' law I(\theta)=I_0\cos^2(\theta-\theta_0). If systematic deviation is observed in the curve, it would indicate the existence of asymmetric secondary polarization structures in the polarization state of entangled photons, providing an alternative explanatory pathway for Bell inequality violation distinct from nonlocality. This experiment employs a high-precision coincidence measurement scheme to ensure statistical significance and designs multiple control experiments to eliminate systematic errors. The experiment can be implemented in laboratories equipped with standard quantum optics equipment.
Malus’s law and the correlations of Bell-type experiments yield the same curve, the squared cosine of the angular offset between two polarizers. We propose a reinterpretation of both phenomena through a single mechanism: the correlation of responses at two points a and b, in series for Malus on a single photon, in parallel for Bell on two identical photons. No other situation is analyzed. The proposed resolution is strictly local, and reassigns cos²(a−b), observed in both cases, as the universal angular ratio of photons. The two experiments thus bring this correlation to light not because paired or polarized photons carry it, but because they make an otherwise invisible correlation testable. Bell’s theorem does not apply to this proposal, because cos²(a−b) is no longer read as the statistics of the responses at a and at b taken separately, but as the statistics of the responses at a and b taken jointly, independently of every other point. In other words, the cos² distribution of responses around an angle a holds only in association with the considered response at a, and does not constrain what the individual statistics of the response at b must be when the response at a is unknown. The article concludes that a strictly local explanation of Bell is possible again, and offers a falsifiable prediction: if identity sharing exists, understanding what it consists of will make it possible to produce identical photons without a common source.
Malus's law and the correlations of Bell-type experiments yield the same curve, the squared cosine of the angular gap between two polarizers. We show that they are the same phenomenon. A polarizer is a pair of orthogonal planes: its response is binary, exact on the axis and on the orthogonal, and the reading of a radial quantity by a plane is a projection, a cosine. A correlation between two polarizers is a double event: one projection factor per verdict, hence the square. The photon contributes only one property, x, defined by a single clause: two photons sharing x respond identically at equal angles. Malus tests x in series, on one photon that conserves it. Bell tests it in parallel, on a pair that shares it. No further hypothesis is required. The squared cosine thus belongs to the polarizer, and the correlation of Bell-type experiments receives a purely mechanical explanation. preprint submitted for peer review additional analysis : https://doi.org/10.5281/zenodo.23630357
Abstract Quantum entanglement is an important action in quantum mechanics. Basically, quantum entanglement of correlated photon pairs can be produced by spontaneous down-conversion process inside the two birefringence crystals. This work aims to create a pair of photons, i.e. signal and idler, that are entangled and to assay the relation between these photons by using polarization-entangled photon pairs to demonstrate quantum non-locality by comparing with the Malus’s law. From the experiments, the coincidence counts as a function of relative angle (α - β) between transmission axis of the polarizer and analyzer were obtained and the polarization entanglement curve was demonstrated. This result corresponding with polarization entanglement prediction term of ½ cos2 (α - β) which confirmed the entanglement of photons.
ABSTRACT This work presents a revised analysis and corresponding perspective of Bell‐type inequalities and their connection to entanglement in polarization experiments with single‐photon resolution. It is argued that standard derivations of Bell‐type inequalities implicitly rely on additional assumptions beyond those originally stated, specifically concerning the cardinalities of the sets of measurements and the sets of photon‐pair properties (Einstein's elements of physical reality, Bell's ). Any model‐based averages must depend on these cardinalities and violations of Bell‐type inequalities become possible for certain ranges of them. In addition, we argue that for Aspect‐type experiments, the statistical properties associated with Bell's (hidden variable) λ can be different before and after actual photon‐transmission through the polarizers. This distinction is particularly necessary for the actual equipment used in the Aspect‐type experiments. Building on these observations, we construct a model based on a consistent evaluation of the measurement events relative to each other. The model reproduces all quantum‐mechanical expectation values for polarization correlations without invoking instantaneous influences at a distance, and additionally yields explicit outcomes for individual photon‐pair measurements. Within this framework, our Einstein‐type Global Space‐Time Model (GSTM) accounts also for polarization correlations in Malus‐type experiments and addresses all commonly discussed paradoxes associated with single‐photon measurements.