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Progress of Theoretical Physics Supplement

Nonlocal Theory of Radiation-Matter Interaction: Boundary-Condition-Less Treatment of Maxwell Equations

K. Cho

Published May 16, 2013• 2 citations
Publisher

Abstract

Based on the separability with respect to position variables of the susceptibility for a finite system, we have developed a formulation which solves Schri:idinger and Maxwell equations selfconsistently.The case of linear response is' described in detail.As a manifestation of the selfconsistency between the two equations, the appearance of radiative damping rate with correct magnitude has been demonstrated for a two-level atom in vacuum.§ 1. Introduction

DOI
10.1143/ptp.106.225

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Nonlocal Theory of Radiation-Matter Interaction: Boundary-Condition-Less Treatment of Maxwell Equations

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Based on the separability with respect to position variables of the susceptibility for a finite system, we have developed a formulation which solves Schri:idinger and Maxwell equations selfconsistently. The case of linear response is' described in detail. As a manifes­ tation of the selfconsistency between the two equations, the appearance of radiative damping rate with correct magnitude has been demonstrated for a two-level atom in vacuum. The interaction of electromagnetic (EM) radiation with matter has been studied in various aspects for quite a long time. The relevant phenomena range, in the scale of wavelength, from rf wave (NMR, for example) to gamma ray (Mossbauer effect): They are related with the absorption, reflection, transmission, propagation, refrac­ tion, scattering, diffraction, emission, etc. of EM wave in linear and nonlinear manner. For the theoretical treatment of these phenomena, we need Schrodinger equation for the description of matter and Maxwell equations in classical or quantized form for radiation field. When the Maxwell equations are treated in classical form, this is called semiclassical treatment of radiation-matter interaction. Though it has a certain limitation, the semiclassical treatment can be appliedto most of the phenom­ ena mentioned above. For the treatment of the interaction of EM radiation with solids, it is usual to employ a susceptibility function to describe the solid, and solve the Maxwell equa­ tions containing that susceptibility as an integral kernel. In doing so, we must consider the boundary conditions (BC) arising from the shape of the sample in question. However, we do not consider BC's in treating microscopic systems such as an atom or small molecule. This would mean that the use of BC(s is in general not compulsory, although we are so much accustomed to it. In this respect, there has been a long debated problem of additional boundary condition (ABC) in relation with exciton polaritons in semiconductors or insulators, where the nonlocal nature of the medium allows the existence of two or more eigenmodes of the coupled exciton-EM radiation. After the long history of the ABC problem,r>- 4 > we now have a recipe to derive the appropriate form(s) of ABC(s) from a microscopic model of the medium including the details of the surface. 5 >-s> Another form of the microscopic solution of the ABC problem has been given as an ABC-free formulation, 9 >-n> where one needs

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Based on the separability with respect to position variables of the susceptibility for a finite system, we have developed a formulation which solves Schri:idinger and Maxwell equations selfconsistently.The case of linear response is' described in detail.As a manifestation of the selfconsistency between the two equations, the appearance of radiative damping rate with correct magnitude has been demonstrated for a two-level atom in vacuum.§ 1. Introduction
Based on the separability with respect to position variables of the susceptibility for a finite system, we have developed a formulation which solves Schri:idinger and Maxwell equations selfconsistently. The case of linear response is' described in detail. As a manifes­ tation of the selfconsistency between the two equations, the appearance of radiative damping rate with correct magnitude has been demonstrated for a two-level atom in vacuum. The interaction of electromagnetic (EM) radiation with matter has been studied in various aspects for quite a long time. The relevant phenomena range, in the scale of wavelength, from rf wave (NMR, for example) to gamma ray (Mossbauer effect): They are related with the absorption, reflection, transmission, propagation, refrac­ tion, scattering, diffraction, emission, etc. of EM wave in linear and nonlinear manner. For the theoretical treatment of these phenomena, we need Schrodinger equation for the description of matter and Maxwell equations in classical or quantized form for radiation field. When the Maxwell equations are treated in classical form, this is called semiclassical treatment of radiation-matter interaction. Though it has a certain limitation, the semiclassical treatment can be appliedto most of the phenom­ ena mentioned above. For the treatment of the interaction of EM radiation with solids, it is usual to employ a susceptibility function to describe the solid, and solve the Maxwell equa­ tions containing that susceptibility as an integral kernel. In doing so, we must consider the boundary conditions (BC) arising from the shape of the sample in question. However, we do not consider BC's in treating microscopic systems such as an atom or small molecule. This would mean that the use of BC(s is in general not compulsory, although we are so much accustomed to it. In this respect, there has been a long debated problem of additional boundary condition (ABC) in relation with exciton polaritons in semiconductors or insulators, where the nonlocal nature of the medium allows the existence of two or more eigenmodes of the coupled exciton-EM radiation. After the long history of the ABC problem,r>- 4 > we now have a recipe to derive the appropriate form(s) of ABC(s) from a microscopic model of the medium including the details of the surface. 5 >-s> Another form of the microscopic solution of the ABC problem has been given as an ABC-free formulation, 9 >-n> where one needs
A complete and detailed treatment of a three-level atom interacting with two near-resonant monochromatic fields is presented. It is assumed that the only damping mechanism is radiative damping, and that the atoms all have the same resonance frequencies, as is the case in an atomic beam. Detailed analytic solutions as well as numerical examples are determined from quantum-electrodynamic equations of motion. In addition, approximation techniques are presented which allow one to get accurate quantitative predictions for strong applied fields from simple rate-equation-like arguments. Absorption and emission spectra are determined, as are the transient and steady-state response of the atom. Two of the more interesting predictions are an emission spectrum containing up to seven components and a prediction of steady-state populations larger than 0.5 in the second excited state.
Abstract To describe the interaction of a two-level atom with radiation we shall use a semiclassical treatment, i.e. the radiation is treated as a classical electric field but we use quantum mechanics to treat the atom. From the theory of the interaction with radiation, we will be able to find the conditions for which the equations reduce to a set of rate equations that describe the populations of the atomic energy levels (with a steady-state solution). In particular, for an atom illuminated by broad band radiation, this approach allows us to make a connection with Einstein’s treatment of radiation that was presented in Chapter 1; we shall find the Einstein B coefficient in terms of the matrix element for the transition.
We investigate the long-time behavior of a model consisting of N two-level atoms in a lossless cavity. The Hamiltonian of our system contains the radiation oscillators in addition to the matter Hamiltonian and the usual ∫j·Adv interaction term. In order to treat the system perturbatively, it would be necessary to remove the tremendous degeneracy of the system. Since this is prohibitively difficult, and since we are interested in the long-time behavior of the system, we solve the quantum mechanical Liouville equation directly for a wide class of physically important initial distribution functions. We show the effective expansion parameter is γ̃Nγ̃ where γ̃ is a dimensionless atomic dipole moment and N is the number of atoms. In the lowest order we find the self-consistent field approximation. In the next order, particle-field correlations appear. We explicitly solve the equations of motion for the particle-field correlations in terms of the average quantities that appear in the self-consistent field approximation. We show the self-consistent field approximation consists of five first-order differential equations. Next we show the equations of motion for the density matrix of the system correct to order (γ̃Nγ̃)2 are equivalent to eight first-order differential equations. The three additional equations are needed to describe the three second moments of the density matrix of the electromagnetic field that appear in second order. Our lowest-order microscopic equations are equivalent to semiphenomenological theories and our higher-order equations contain only the measurable second-order moments of the electromagnetic field in addition to the variables that appear in semi-phenomenological theories.
This paper presents a realistic model that describes radiation-matter interactions. This is achieved by a generalization of first quantization, where the Maxwell equations are interpreted as the electromagnetic component of the Schroedinger equation. This picture is complemented by the consideration of electrons and photons as real particles in three-dimensional space, following guiding conditions derived from the particle-wave-functions to which they are associated. The guiding condition for the electron is taken from Bohmian mechanics, while the photon velocity is defined as the ratio between the Poynting vector and the electromagnetic energy density. The case of many particles is considered, taking into account their statistical properties. The formalism is applied to a two level system, providing a satisfactory description for spontaneous emission, Lamb shift, scattering, absorption, dispersion, resonance fluorescence and vacuum fields. This model adequately describes quantum jumps by the entanglement between the photon and the atomic system and it will prove to be very useful in the simulation of quantum devices for quantum computers and quantum information systems. A possible relativistic generalization is presented, together with its relationship to QED.