International Scientific-Technical Conference on Actual Problems of Electronic Instrument Engineering (APEIE)
Time Evolution of Three-Photon Interference in Young's Experiment Scheme within Numerical Simulation of 6-Component Coordinate Wave Function of Directed Radiation
The time evolution of the interference pattern obtained from numerical simulation of directed radiation to a given point on the screen is demonstrated. This pattern arises from three identical single-photon sources simultaneously emitting photons in a thought experiment of the Young's experiment type. The simulation uses a 6-component coordinate wave function of a photon – a wave packet as a superposition (integral) of generalized eigenfunctions of the photon's energy, momentum, and helicity operators. This wave function is determined by only two free parameters: the wavelength and the emission time of each photon, understood as the average time interval during which a photon can be emitted by a single-photon source. Calculations were performed for a wavelength of 0.05 cm, emission time of 4 ps, distance between radiation sources of 0.25 cm, and distance from sources to the screen of 0.905 cm. As a result, the interference pattern is established for six consecutive time moments from 29.2 to 34.2 ps, covering the most interesting region of its dynamics. A comparison is made between the interference pattern and similar results previously obtained by numerical simulation of two-photon interference under the same radiation parameters and Young's experiment geometry. It is emphasized that the performed simulation of three-photon interference can find application for the theoretical description of possible interference processes accompanying the implementation of quantum computations based on qutrits as basic elements of quantum computers.
The time evolution of the interference pattern obtained from numerical simulation of directed radiation to a given point on the screen is demonstrated. This pattern arises from three identical single-photon sources simultaneously emitting photons in a thought experiment of the Young's experiment type. The simulation uses a 6-component coordinate wave function of a photon – a wave packet as a superposition (integral) of generalized eigenfunctions of the photon's energy, momentum, and helicity operators. This wave function is determined by only two free parameters: the wavelength and the emission time of each photon, understood as the average time interval during which a photon can be emitted by a single-photon source. Calculations were performed for a wavelength of 0.05 cm, emission time of 4 ps, distance between radiation sources of 0.25 cm, and distance from sources to the screen of 0.905 cm. As a result, the interference pattern is established for six consecutive time moments from 29.2 to 34.2 ps, covering the most interesting region of its dynamics. A comparison is made between the interference pattern and similar results previously obtained by numerical simulation of two-photon interference under the same radiation parameters and Young's experiment geometry. It is emphasized that the performed simulation of three-photon interference can find application for the theoretical description of possible interference processes accompanying the implementation of quantum computations based on qutrits as basic elements of quantum computers.
The formation of an obvious interference pattern in the distribution of the probability of the simultaneous hit of two alone photons radiated synchronously from two independent single-photon sources, playing the slits role of Young's experiment first screen, at different points of the second screen, is demonstrated by modeling the photon coordinate wave function. This function is single-photon wave packet built within the framework of the developed quantum mechanics of the photon in previous authors works, which expresses a superposition of six-component “plane waves with circular polarization” - the eigenfunctions of the operators of photon momentum, of its energy and helicity. The results received make it eventual to justify the description of one-and two-photon interference in framework of “primary” quantization of the electromagnetic field, without resorting to the probability amplitudes of quantum field theory, which, in turn, makes it possible to develop a metaphysical interpretation of experiments with single photons, such as quantum computing, quantum teleportation, quantum cryptography. It seems that this interpretation can serve as an impetus for the development of new practical devices and the setting up of new experiments to reveal the physical nature of quantum interactions through the exchange of photons and, in particular, the physical essence displayed by the idea of quantum object wave function.
Single-photon interference in Young's mental experiment is explained using an approximate 6-component photon wave function in coordinate representation, describing spherically symmetric emission from a single-photon source. This interference is modeled by calculating the photon detection probability density at the corresponding point on the second screen at four selected time moments. It is illustrated for emission with wavelength of 633 nm, an emission duration of 0.667 ns per photon, a slit separation of 3 µm, and a distance of 10 cm from the slits to the second screen where interference is "observed." In addition to the purely quantum mechanical approximate 6-component coordinate photon wave function, a corresponding 1-component "quasi-classical" photon wave function is also used. The resulting interference pattern, which perfectly matches for both functions, suggests that single-photon interference phenomena can be adequately described within the framework of "primary" quantization of the electromagnetic field. It is emphasized that the traditional use of the essentially postulated Feynman transition amplitude to explain Young's experiment with individual particles is justified by the considered 1-component coordinate photon wave function (along with the 6-component quantum mechanical one), providing a richer description of interference not just in arbitrary units, but in units of calculable probability density, which can also be traced in terms of its evolution in space and time. Attention is also drawn to the fact that "measurable" (in some recent experiments) coordinate photon wave functions can only be associated with the 1-component quasi-classical photon wave functions, which are studied in detail in this article.
In modern areas of photonics, the physical description of the interaction of photons with matter in the control, transmission and registration of single-photon and two-photon states implemented in practice is of great importance. An appropriate acceptable description may be faced with the need to take into account various kinds of interference effects associated with these states. Meanwhile, even the most “simple” case of single-photon interference in Young's experiment requires the use of a rather complex apparatus of quantum electrodynamics. This article explains one- and two-photon interference in Young's thought experiment based on the photon wave function (PWF) in coordinate representation. This explanation is illustrated by two examples of wavelengths: 10.6 μm and 1.5 cm. For both examples, two approaches to PWF modeling are used: “purely quantum-mechanical” and “quasi-classical”. In the first approach, a 6-component coordinate PWF is constructed using a spherically symmetric momentum distribution in a wave packet, followed by numerical and approximate analytical calculations. In the second approach, a one-component “quasi-classical” PWF is constructed, which corresponds to either electric dipole radiation or simulated spherically symmetric radiation. In all cases, the same pronounced interference pattern was obtained, which allows us to conclude that not only the quantum-mechanical coordinate PWF is able to explain the phenomena of one- and two-photon interference, but also a much simpler “quasi-classical” PWF. This conclusion sheds light on the theoretical interpretation of the measurement of the coordinate PWF in some recent experiments.
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.
The description of photon-matter interaction upon control, transmission, and detection of single-photon, two-photon, and multiphoton states, including the entangled ones, will play an ever-increasing role in many areas of photonics. An appropriate description requires taking into consideration various types of interference effects associated with these states. However, the relatively complex apparatus of second quantization of the electromagnetic field is used even in the simplest single-photon experiments equivalent to the Young’s one, e.g., the experiments with the Mach–Zehnder interferometer. In the present work, the Young’s single-photon mental experiment is explained using the coordinate photon wave function (PWF). The explanation is illustrated by specific examples of the single-photon states at certain wavelengths and different duration of radiation within the framework of two approaches: the quantum mechanical and the “quasi-classical.” In the first approach, a 6-component coordinate PWF is constructed based on the spherically symmetric momentum distribution in a wave packet, followed by approximate analytical calculations. In the second approach, a one-component quasi-classical PWF corresponding to the electric-dipole radiation is constructed. The same pronounced interference pattern was obtained in both cases, which makes is possible to draw the conclusion that the coordinate PWF allows explaining the one- and two-photon interference phenomena. This conclusion sheds the light on theoretical interpretation of the measurement of the coordinate PWF carried out in some of the recent experiments.
The main principles of photon quantum mechanics describing its one-particle states with the help of the wave function in coordinate representations are given.This wave function (packet) is the superposition of the basic bivectors that are generalized eigenfunctions of energy, momentum, and helicity operators.The quantum-mechanical approach is offered for an explanation of the interference Young's experiment.This explanation is especially important for new interpretation of the non-laser method of obtaining the interference by the amplitude division, as in this case it is traditionally assumed that the radiation of one light train of a single atom interferes with itself.In our explanation, not both "halves" of some real train ("scrap") of the electromagnetic wave interfere with each other, but the both terms of the photon's wave function do this.Nevertheless, the wave function is not directly measured by experience, and therefore it does not exist as a physical object.The attraction of the wave function to the explanation of single-photon interference obtained "by the division of the wave front" is obviously also necessary.Therefore, the situation for the photons emitted by individual atoms (and also, obviously, by laser), is absolutely similar to the situation with the particles having mass, whose distribution in space is described by wave function in coordinate representation.It is pointed out that in fact the photon is not some "formed" quantum particle, but it is a quasi-particle arising when a certain spin wave propagates in a physical vacuum at Planck distances.