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Journal of Physics Conference Series

Interaction of broadband quantum fields with resonant atoms in second-order algebraic perturbation theory

Ashat M. Basharov

Published Aug 1, 2018• 0 citations
Publisher

Abstract

The second-order algebraic perturbation theory provides a correct and conventional description of the interaction of a single-mode quantized field and a broadband field with resonant particles, and in case of interacting two different broadband fields in the Markov approximation the generalization of the Hudson-Parthasarathy algebra is required to derive the Schrödinger equation.

DOI
10.1088/1742-6596/1068/1/012003

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The second-order algebraic perturbation theory provides a correct and conventional description of the interaction of a single-mode quantized field and a broadband field with resonant particles, and in case of interacting two different broadband fields in the Markov approximation the generalization of the Hudson-Parthasarathy algebra is required to derive the Schrödinger equation.
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A large number of "molecules," which are taken to be angular momentum oscillators of frequency $\ensuremath{\nu}$, are considered to interact through their dipole moment with two electromagnetic cavity modes of frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$, respectively, where $\ensuremath{\nu}={\ensuremath{\omega}}_{1}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\omega}}_{2}$. The behavior of the combined system is analyzed up to fourth order in perturbation theory. Phenomena known as two-photon emission or absorption, parametric amplification, various types of Raman emission, and frequency conversion are exhibited. The analytical method used is applicable to both a quantum-mechanical and a classical description of the modes and molecules; it displays explicitly the resulting differences. Expressions for the rate of energy change of the modes and molecules indicate resonance in second and fourth order. The second-order results are the same for the classical and quantum-mechanical treatments, while the fourth-order results show certain differences, which are examined in detail. The conditions under which each of the above phenomena occurs are discussed in a self-consistent manner, both classically and quantum-mechanically. It is shown that the second-order expression and the most significant part of the fourth-order expression can be derived very simply from a resonance interaction Hamiltonian that consists of a sum of products of three variables---two referring to the two modes, and the third to the molecules. The relationship between this resonance Hamiltonian and the fundamental dipole-moment Hamiltonian is examined, and the approximations by which the resonance Hamiltonian may be derived are considered. Arguments concerning the applicability of perturbation theory to steady-state situations are presented; it is pointed out that the second-order expressions for the energy transfer between molecules and field may be applicable to such a situation, in which case the classical and quantum-mechanical treatments are equivalent.
Abstract The algebraic resonance perturbation theory (ARPT) is formulated, general expressions of the theory are written down, by means of which effective Hamiltonians are concisely derived, interference interactions are described, and kinetic equations are obtained for various cases of interaction between electromagnetic radiation and quantum systems. The natural integration of the method of stochastic differential equations into ARPT and the physical justification of the principles for the selection of ARPT terms are indicated in the case of approximating the environment of an open system by delta-correlated noise. The inevitability of the emergence of interference interactions is emphasized due to the formulated ARPT requirements regarding the selection of only slowly time-varying terms in the Dirac picture into the effective Hamiltonian.
Abstract Quantum stochastic differential equations are shown to naturally arise because of the mathematical background of the correctness of the Schrödinger equation, which describes the interaction of an open quantum system with broadband quantized fields in the Markov approximation. The main classical and quantum stochastic processes appear in the notation in the Markov approximation of the operator of the electric broadband quantized field strength and the effective Hamiltonian of the interaction of a quantum system with it obtained by the methods of algebraic resonance perturbation theory. All basic stochastic quantum processes, including the counting and two-quantum Lévy processes, are constructed based on quantum white noise, which differs significantly from the classical case. At the same time, the apparatus of stochastic differential equations demonstrates consistency and parallelism.
The algebraic approach to operator perturbation method has been applied to two quantum--mechanical systems ``The Stark Effect in the Harmonic Oscillator'' and ``The Generalized Zeeman Effect''. To that end, two realizations of the superoperators involved in the formalism have been carried out. The first of them has been based on the Heisenberg--Dirac algebra of a^†\hat{a}^\daggera^†, a^\hat{a}a^, 1^\hat{1}1^ operators, the second one has been based in the angular momemtum algebra of L^+\hat{L}_+L^+​, L^−\hat{L}_-L^−​ and L^0\hat{L}_0L^0​ operators. The successful results achieved in predicting the discrete spectra of both systems have put in evidence the reliability and accuracy of the theory.
The classical or semiclassical theory presented in Chapter 2 gives a satisfactory account of light-matter interactions in some cases. However, a detailed understanding of different processes requires a quantum mechanical theory of radiation. The quantum theory of light, like classical theory, is also based on Maxwell's equations. However, in quantum theory the electric field F and the magnetic field H are treated as operators. This chapter develops the correspondence between the two and outlines the method for converting field vectors F and H into quantum mechanical operators. Topics discussed include wave equation in a rectangular cavity, quantization of the electric field, time-dependent perturbation theory, interaction of an electron with the magnetic field, and second-order perturbation theory. Exercises are provided at the end of the chapter.