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Light & Engineering

Coherent Stimulated Radiation, Coherent Spontaneous Radiation and Coupled Photon Pairs in Non-Uniform Thermally Excited Gaseous Media: The Theory

B. A. Veklenko

Published Feb 1, 2022• 0 citations
Publisher

Abstract

It is demonstrated that presence of stimulated radiation processes in gaseous media makes it possible to generate rays inversely reflected from interfaces of thermally excited gaseous media and the processes of left-hand refraction of the rays having passed through these media. These processes are caused by availability of coupled photonic states and stimulated radiation in media the properties of which differ from those of stimulated radiation in vacuum. Estimation of these processes is based on exclusion of the procedure of forced breaking of quantum correlators and correct consideration of high-order correlators. Along with Schrödinger’s and Heisenberg’s representations, this article uses also the less-known Γ- representation, which allows us to reduce a multi-particle problem of quantum electrodynamics to a single-particle problem within the an infinite-dimensional space. The equations formed in Г- representation are mathematically simpler and may be studied using the standard Wick’s theorem rather than its approximated thermodynamic variant suggested by T.A. Matsubara. Therefore it is possible to keep all possible solutions of the initial system of equations in mind. Of course, approximation methods are used for explicit solution of these equations. Using Feynman diagrams instead of the Dyson equation for quantum correlators, it is possible to find an equation defining the complete density matrix of a photonic sub-system.

DOI
10.33383/2021-099

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It is demonstrated that presence of stimulated radiation processes in gaseous media makes it possible to generate rays inversely reflected from interfaces of thermally excited gaseous media and the processes of left-hand refraction of the rays having passed through these media. These processes are caused by availability of coupled photonic states and stimulated radiation in media the properties of which differ from those of stimulated radiation in vacuum. Estimation of these processes is based on exclusion of the procedure of forced breaking of quantum correlators and correct consideration of high-order correlators. Along with Schrödinger’s and Heisenberg’s representations, this article uses also the less-known Γ- representation, which allows us to reduce a multi-particle problem of quantum electrodynamics to a single-particle problem within the an infinite-dimensional space. The equations formed in Г- representation are mathematically simpler and may be studied using the standard Wick’s theorem rather than its approximated thermodynamic variant suggested by T.A. Matsubara. Therefore it is possible to keep all possible solutions of the initial system of equations in mind. Of course, approximation methods are used for explicit solution of these equations. Using Feynman diagrams instead of the Dyson equation for quantum correlators, it is possible to find an equation defining the complete density matrix of a photonic sub-system.
This work continues the author’s study of the processes of interaction between a quantum electromagnetic field and a spatially non-uniform gaseous medium [1, 5]. Previously unknown optical effects caused by consideration of high-order quantum correlators were discovered. This is caused by the problem of calculation of high-order photonic correlators starting from the fourth order, which appears when the interaction between the quantum electromagnetic field is described using the Feynman diagrams, According to T.A. Matsubara’s technique applied nowadays, these correlators are just neglected. The author’s articles [1, 5] describe the calculation method allowing us to avoid thiscomplication. The method is based on the following. High-order quantum correlators appear when multi-particle problems are studied. If the number of particles is small (hydrogen atom), the problem disappears. Therefore, if we replace a multi-particle problem with a single-particle problem (even if this particle is rather complex), the problem will be solved. Such solution (by using the Γ-operators technique) is described in the said author’s articles. While the standard secondary quantisation technique uses operators of creation and destruction of separate particles, the Γ-operators technique deals with operators of creation and destruction of photon conglomerates (systems). It turns out that high-order correlators just do not appear when the Γ-operators technique is used. This article demonstrates a specific example of application of the Γ-technique.
According to quantum electrodynamics, the cross section for resonant scattering of radiation on an aggregate of excited atoms can be written as a sum of positive definite terms. This type of structure is not consistent with the Fresnel formulas for the reflection coefficient of radiation from thermally excited media. The difference shows up on a macroscopic level and indicates that semiclassical radiation theory cannot be used. A study of the correlation between elastic scattering and stimulated emission processes clarifies the reason for the discrepancies. The resulting singularities require summing of Feynman diagrams which appear beginning in the sixth order of perturbation theory. A lower bound estimate for the reflection coefficient from a plane layer is given, including processes which violate the statistics of radiation. The contribution of stimulated emission processes caused by the initially scattered photon are examined specifically. An experiment is proposed which would settle the choice of theories.
A new method for calculating the density matrix of a quantized electromagnetic field which interacts with the environment in the presence of kinetic processes in the medium is suggested. This method accurately takes into account photon-photon quantum correlators and possesses certain symmetry properties, which substantially facilitate the summation of the Feynman diagrams that arise in the calculations. Forward scattering of resonance radiation by two-level gas atoms in a magnetic field is considered as an application. Insufficiency of a semiclassical description of this coherent process in excited media with the use of the unique tensor index of refraction is shown to follow from quantum electrodynamics. The introduction of one more function depending on the frequency of irradiation and the concentration of excited atoms is necessary. Experiments that simultaneously determine both this additional function and the standard refractive index of the medium are specified.
The quantum fluctuations of an electromagnetic field in thermally excited media are calculated using the quantum electrodynamical method of Γ operators and without invoking phenomenological elements. The drawbacks of the standard theory based on the fluctuation-dissipation theorem and on the Matsubara technique of temperature Green’s functions are indicated. The decisive role of the correct consideration of higher order photon-photon correlators and the inadmissibility of their approximation by products of lower order correlators are underlined. It is shown that, contrary to the accepted opinion, the quantum fluctuations of an electromagnetic field cannot be expressed via the refractive indices of media introduced into the theory for the calculation of mean fields. The results obtained are compared with those previously derived using the Matsubara technique for calculation of the Green’s functions of an electromagnetic field in dispersive media under conditions of thermodynamic equilibrium. The previous results are shown to be incorrect at least for media consisting of atoms or molecules with a discrete energy spectrum.
Recently Glauber has described the properties of coherent radiation fields, and has constructed the density matrix of the field in two simple cases: (1) The radiating system is a classical radiator and no reaction is considered; (2) the central-limit theorem applies to a collection of radiators. This paper investigates other simple "almost" exactly soluble problems, in which a quantum-mechanical two-level system interacts with a quantized electromagnetic field originally in a pure coherent state in a single mode. The first-order correlation function ${G}^{(1)}=〈{E}^{\ensuremath{-}}{E}^{+}〉$ is compared with $〈{E}^{\ensuremath{-}}〉〈{E}^{+}〉$ at resonance when the stimulating field is initially a pure coherent state and the two-level system is initially in its excited state. The corresponding quantities are also computed for a field whose initial density matrix is a Gaussian superposition of coherent states (e.g., blackbody radiation), as well as for a field which is initially described as having a given number of photons.
A model of radiation by multi-atoms is discussed. System A, composed of N atoms with two energy levels, is coupled with a single-mode radiation B. A and B are in turn coupled with two thermal reservoirs D and C respectively, and we also assume that all atoms which are located within a wave length of the radiation field have the same phase. Then under the approximation of long wave, rotating wave and mean fields, by introducing Markov approximation after using the method of density matrix, we have derived the of equations motion for systems A and B respectively. These equations are concerning with the following processes: the spontaneous emission of single atom, the absorption and induced emission stimulated by thermal radiation, the induced emission of N atoms stimulated by single mode coherent state radiation as well as the coherent spontaneous emission of this multi-atoms system. The latter two processes are actually the two kinds of superradiance originated from Dicke model and had been investigated previonsly. We have analysed these five processes by numerical calculations. Besides, the influence of these processes upon the evolution of atomic system attaining thermal equilibrium and the increase of entropy is also discussed. It is pointed out that the correlations inside a multi-atoms system would not be destroyed by coherent spontaneous emission, in contrast with the incoherent one which leads to the change of entropy in atomic system. However, only when the entropy of the two reservoirs has been taken into account, will it be possible to demonstrate the mono-tonic increase in total entropy.