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Инженерная физика.

ON THE COORDINATE WAVE FUNCTION OF TWO PHOTONS IN ENTANGLED STATE AT THE OUTPUT OF NONLINEAR CRYSTAL

А.П. ДАВЫДОВ, Tatiana Pavlovna Zlydneva

Published Dec 22, 2021• 0 citations
Publisher

Abstract

Конструируется волновая функция в координатном представлении двух запутанных фотонов в состоянии на выходе из нелинейного кристалла путем моделирования векторов поляризации и применения гауссовского импульсного распределения для каждого из фотонов. The wave function is constructed in the coordinate representation of two entangled photons in a state at the exit from a nonlinear crystal by modeling the polarization vectors and applying a Gaussian momentum distribution for each of the photons.

DOI
10.25791/infizik.12.2021.1242

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Конструируется волновая функция в координатном представлении двух запутанных фотонов в состоянии на выходе из нелинейного кристалла путем моделирования векторов поляризации и применения гауссовского импульсного распределения для каждого из фотонов. The wave function is constructed in the coordinate representation of two entangled photons in a state at the exit from a nonlinear crystal by modeling the polarization vectors and applying a Gaussian momentum distribution for each of the photons.
Abstract The main provisions of the construction of the single-particle wave function of a photon in the coordinate representation are expounded and its application to the description of single-photon interference phenomena is discussed. The connection between the concepts of electromagnetic field strengths used in quantum electrodynamics and the characteristics of wave packets in photon quantum mechanics is substantiated. For illustration, we consider the relationship between the structure of the photon wave function in coordinate representation and the polarization of photon in the formalism of quantum transition amplitudes. The meaning of the concept of linear polarization of a photon in the state of a wave packet is explained. The wave function is constructed in the coordinate representation of the state of two entangled photons at the exit from a nonlinear crystal by modeling the polarization vectors and Gaussian momentum distribution for each of the photons.
С помощью моделирования волновой функции фотона в координатном представлении демонстрируется возникновение отчетливой интерференции в вероятности попадания двух одиночных фотонов в разные точки второго экрана в схеме опыта Юнга, если эти фотоны одновременно испускаются независимыми источниками. By modeling the photon wave function in the coordinate representation, the emergence of distinct interference in the probability density of hitting of the two single photons into different points of the second screen in scheme of Young’s experiment is demonstrated, if these photons were simultaneously emitted by independent sources.
Объясняется одно- и двухфотонный опыт Юнга с помощью 6-компонентной квантовой и 1-компонентной «квазиклассической» волновых функций фотона в координатном представлении, которые моделируются различными механизмами дифракции фотона на отверстиях или одновременного излучения двух фотонов (двумя) независимыми источниками. Это объяснение сопоставляется с хорошо известным вариантом объяснения интерференции частицы «сама с собой» при ее прохождении «сразу через оба отверстия» в опыте Юнга с помощью постулированных амплитуд Фейнмана. Утверждается, что эти амплитуды не могут быть волновыми функциями, в прямом смысле, описывающими некоторые состояния квантовых частиц. Во всех случаях, в предположении линейной поляризации фотонов, получена схожая ярко выраженная интерференционная картина, позволяющая сделать вывод, что координатная волновая функция фотона способна объяснить явления одно- и двухфотонной интерференции. В свою очередь, этот вывод дает возможность ассоциировать измеряемую координатную волновую функцию фотона (и двух фотонов в запутанном состоянии), в некоторых недавних экспериментах, с «квазиклассической» волновой функцией фотона (и двух фотонов), хотя она и не учитывает его (их) поляризационных свойств. Young's single- and two-photon experiments are explained using 6-component quantum and 1-component "quasi-classical" photon wave functions in coordinate representation, which are modeled by various mechanisms ofphoton diffraction on holes or types of simultaneous emission of two photons by (two) independent sources. This explanation is compared with the well-known version of the explanation of the interference of a particle "with itself when it passes "through both holes at once" in Young's experiment using Feynman's postulated amplitudes. It is argued that these amplitudes cannot be wave functions, in the literal sense, describing some states of quantum particles. In all cases, under the assumption of linear polarization of photons, a pronounced interference pattern was obtained, allowing us to conclude that the coordinate wave function of the photon is capable of explaining the phenomena of single- and two-photon interference. In turn, this conclusion makes it possible to associate the coordinate wave function of the photon (and two photons in an entangled state), measured in some recent experiments, with the "quasi-classical" wave function of the photon (and two photons), although it does not take into account its (their) polarization properties.
Introduction: In the rapidly developing areas of photonics associated with quantum teleportation of photons, quantum cryptography, and quantum computing, ideas are becoming increasingly relevant about the certain localized states of single or several entangled photons moving in space and time. As is known, in quantum mechanics, localized states of particles can only be described using wave functions in coordinate representation – the corresponding wave packets. Aims: To explain Young's one- and two-photon experiments using coordinate 6-component quantum mechanical and 1-component quasi-classical photon wave functions. Methods: The article outlines the method developed in authors' previous work in constructing the 6-component coordinate wave function of free photons in the form of the wave packet (an integral over the entire momentum space and sum over two possible values +1 and -1 of the helicity). The basis functions are the circularly polarized monochromatic waves, corresponding to certain values of the momentum, helicity, and photon energy operators and forming a complete orthonormal set of generalized eigenfunctions of these operators. This photon wave function (PWF) is normalized to the unit probability of finding the photon anywhere in space. Both the basis functions and the coordinate PWF satisfy a Schrödinger-type equation derived from the Maxwell-Lorentz equations written in quantum mechanical form first introduced by Majorana. Then the results of modeling and consideration of the space-time evolution of the wave packet with Gaussian momentum distribution corresponding to the directed photon emission for 80 fs are briefly presented. Further, in view of the possibility of constructing the coordinate PWF, the basic formula for the waveparticle duality of photons and particles is formulated. It is emphasized that the photon is fundamentally different from “ordinary” particles, since, according to the authors, the photon is the result of the propagation of a certain complex spin wave in a physical vacuum, the details of which can only be considered simultaneously with the study of the structure of the physical vacuum at Planckian distances. Results: Young`s one- and two-photon experiments are explained using various options for modeling 6-component coordinate PWFs, including approximate functions corresponding to spherically symmetric and electric dipole photon radiation. Conclusion: This explanation is illustrated using specific examples of radiation and, at the same time, is compared with numerical simulations for spherically symmetric radiation and the use of 1-component quasi-classical PWFs.
Logical analysis of the phenomenon of entanglement in terms of quantum theory leads to the conclusion that the anticipating measurement of entangled photons in one channel, leads to the creation in the second channel a mixture of the two plane-polarized flows with mutually perpendicular polarizations. Путенихин П.В. Логический анализ явления запутанности с точки зрения квантовой теории приводит к выводу, что опережающее измерение запутанных фотонов в одном канале приводит к созданию во втором канале смеси двух плоскополяризованных потоков с взаимно перпендикулярными поляризациями.