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Journal of physics. A, Proceedings of the Physical Society. General

A quantum theoretic treatment of double-resonance phenomena

Daniel F. Walls

Published Sep 1, 1971• 9 citations
Publisher

Abstract

The author presents a quantum mechanical method capable of providing exact non-perturbative solutions in problems involving the interaction of atoms and molecules with the electromagnetic field. To illustrate the method two examples of double-resonance phenomena are considered. In the first example the author studies the interaction of a near-resonance rf field with the Zeeman sublevels of atomic hydrogen. In the second the interaction of a three-level molecule with two applied electromagnetic fields is studied. Both the molecular levels and the electromagnetic field are quantized. The method eliminates the deficiencies of a semiclassical treatment, yet does not require the sophisticated computations of the resolvant formalism of Cohen-Tannoudji.

DOI
10.1088/0305-4470/4/5/005

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The author presents a quantum mechanical method capable of providing exact non-perturbative solutions in problems involving the interaction of atoms and molecules with the electromagnetic field. To illustrate the method two examples of double-resonance phenomena are considered. In the first example the author studies the interaction of a near-resonance rf field with the Zeeman sublevels of atomic hydrogen. In the second the interaction of a three-level molecule with two applied electromagnetic fields is studied. Both the molecular levels and the electromagnetic field are quantized. The method eliminates the deficiencies of a semiclassical treatment, yet does not require the sophisticated computations of the resolvant formalism of Cohen-Tannoudji.
One of the simplest nontrivial problems involving the atom–field interaction is the coupling of a two-level atom with a single mode of the electromagnetic field. A two-level atom description is valid if the two atomic levels involved are resonant or nearly resonant with the driving field, while all other levels are highly detuned. Under certain realistic approximations, it is possible to reduce this problem to a form which can be solved exactly; allowing essential features of the atom-field interaction to be extracted. In this chapter we present a semiclassical theory of the interaction of a single two-level atom with a single mode of the field in which the atom is treated as a quantum two-level system and the field is treated classically. A fully quantum mechanical theory will be presented in Chapter 6. A two-level atom is formally analogous to a spin-1/2 system with two possible states. In the dipole approximation, when the field wavelength is larger than the atomic size, the atom–field interaction problem is mathematically equivalent to a spin-1/2 particle interacting with a time-dependent magnetic field. Just as the spin-1/2 system undergoes the so-called Rabi oscillations between the spin-up and spin-down states under the action of an oscillating magnetic field, the two-level atom also undergoes optical Rabi oscillations under the action of the driving electromagnetic field. These oscillations are damped if the atomic levels decay. An understanding of this simple model of the atom–field interaction enables us to consider more complicated problems involving an ensemble of atoms interacting with the field.
Abstract The interaction of three and two-level atomic systems with two electromagnetic fields, a weak near resonant high-frequency and a strong off-resonant low-frequency ones, is analyzed by applying the method of multiple-scale perturbation theory. We consider two examples: (1) a three-level Λ system interacting with a weak XUV field at the high-energy transition and a strong IR field at the adjacent low-energy transition. The IR field will periodically Stark modulate the energy of the most excited state, which will result in generation of sidebands of the XUV field at the frequencies multiple of the IR field modulation frequency; (2) a two-level atom interacting with a weak near-resonant high-frequency and an off-resonant low-frequency field, which is related to the quantum amplification by superradiant emission of radiation problem.
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.
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.
The interaction of a quantum system with an oscillating field is studied in a formalism which replaces the semiclassical time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The formalism is developed as a mathematical equivalent to the semiclassical treatment, and interpreted as a classical approximation to the quantum treatment of the field. Combined with a perturbation theory for two nearly degenerate states, the formalism provides a convenient method for determining resonance transition probabilities including frequency shifts and multiple quantum transitions. The theory is illustrated by a detailed study of the simple case of a two-state system excited by a strong oscillating field.
We investigate in this chapter the general problem of the interaction of an atom (or a molecule) and a classical electromagnetic field. Its importance derives firstly from the fact that a large part of our knowledge of atoms is obtained from a study of the radiation they absorb or emit (we shall consider equally visible radiation and radio-frequency and X-rays) and, secondly, that the interaction with matter modifies the propagation of the electromagnetic field itself, notably through absorption, refraction or scattering. The interactions of atoms with light therefore encompass a vast range of physical effects that we could not hope to cover adequately in a single chapter. In this chapter we shall therefore present the fundamental features of the interaction of an atom, which will be treated quantum mechanically, with a classical electromagnetic field, that is an electromagnetic field described by real electric and magnetic vectors obeying Maxwell's equations. A rigorous description of the atom–light interaction would have to take into account the quantum nature of light. This we shall leave to Chapter 6. It turns out, nevertheless, that many important results can be obtained from the semi-classical viewpoint adopted here (although it is perhaps more appropriate to refer to this treatment as ‘semi-quantized’).