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Canadian Journal of Physics

A reexamination of the Green function treatment of the Mollow problem

D. A. Hutchinson

Published Feb 1, 1985• 1 citation
Publisher

Abstract

The application of the Green function method to the Mollow problem is reconsidered. This problem consists of an intense monochromatic electromagnetic field in resonance with a two-level atom. We obtain a solution to the infinite hierarchy of Green function equations of motion instead of decoupling the equations of motion. This solution gives an excitation spectrum in agreement with Mollow's results. We conclude that earlier discrepancies in the excitation spectrum were artifacts of the decoupling procedure. Finally, we comment upon the physical ideas associated with the present and previous Green function treatments of the Mollow problem.

DOI
10.1139/p85-022

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The application of the Green function method to the Mollow problem is reconsidered. This problem consists of an intense monochromatic electromagnetic field in resonance with a two-level atom. We obtain a solution to the infinite hierarchy of Green function equations of motion instead of decoupling the equations of motion. This solution gives an excitation spectrum in agreement with Mollow's results. We conclude that earlier discrepancies in the excitation spectrum were artifacts of the decoupling procedure. Finally, we comment upon the physical ideas associated with the present and previous Green function treatments of the Mollow problem.
The Green function theory is successful in many fields of theoretical physics and the Bogoliubov-Gor'kov theory system is its important branch that involves many theoretical methods. In the last twenty years many papers have been published, but as all the equations of motion are coupled and their number is infinite, its global structure problem has not been investigated as yet. This paper is devoted to the study of its global structure, the exact decoupling problem, and the uniqueness and completeness problems. Some higher order spectral representation theorems and exact relationships between higher and lower order Green functions are obtained. Thus the equations of motion are decoupled exactly. In this paper it is proved that after cutting off the equations of motion are decoupled exactly and the solution of these equation systems may not be unique. It is also proved that if there is one solution satisfying all the Green function's equations, then there must be a solution set, all of them satisfy all the equations and its number is infinite and the differences between them are arbitrary. By adding some limitations to ImG(x) at the real axis a uniqueness theorem is proved.
A Green's function formulation of the problem of resonance fluorescence from a system of identical two level atoms is presented. The general formulation is suitable for the investigation of a variety of nonlinear phenomena, e.g., propagation of electromagnetic waves inside the medium, fluorescence from a finite medium, optical bistability, atomic correlations, and additional side bands. Translational motion, collisions, Coulomb dipole–dipole interaction, excitation transfer, and local field corrections are also included in the basic equations. The effect of atom–atom correlations in the relaxation mechanism (radiative or collisional) is also taken into account. Finally, the results are also presented in the "generalized dressed atom" (GDA) representation. Applications of the general theory will be given in a future paper.
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.
The spectral density of resonance fluorescence from a two-level atom in a monochromatic radiation field of arbitrary strength and detuning is calculated by solution of the wave equation. A basis set in which the strong coupling of the atom and the quantised incident field is diagonal is used, and the resulting infinite set of coupled equations is solved by truncating them to a set of N equations and finally proceeding to the N to infinity limit. The resulting spectral densities of the inelastically scattered field for various levels of intensity of the incident field and degree of detuning are found to be identical with those of Kimble and Mandel (1976, 1977) obtained by solution of the coupled equations of motion of the atom and the radiation field in the Heisenberg picture.
The power spectrum of the light scattered by a three-level atom driven near resonance between the ground state (|1>) and the second excited state (|3>) by a monochromatic classical electric field is evaluated following the method developed by B. R. Mollow. The incident field is assumed to interact with the dipole moment between the states |1> and |3>, while the emitted light is assumed to interact with that between the state |3> and the first excited state |2> of the atom. The atom is assumed to relax to equilibrium only via radiation damping for the sake of simplicity. The power spectrum of the scattered field is explicitly calculated from the two-time atomic dipole moment correlation function, which is evaluated on a Markoff-type assumption, and various limiting cases are discussed. In Appendix the case is considered in which radiative damping between the higher excited state and the ground state is also present.