Theory of resonance fluorescence

The expectation value of any physical observable, represented for instance by the operator, is then given by 

We see that the observable properties of this atom are determined not so much by the individual coefficients a (t) 

In resonance fluorescence experiments, however, we are never able to study just a single atom, rather we are forced to investigate the properties of a sample containing N atoms. The measurable properties of this sample are then given in terms of the average values,, of a set of physical observables taken over the ensemble of independent atoms where 

For this reason it is now convenient to use the average values of the products of the probability amplitudes to define 

From equations (15.9) and (15.10) we see that the mean value of the observable can be expressed in terms of the ensemble density matrix by 

---

Corney, A. and Series, G.W. (1964). Theory of resonance fluorescence excited by modulated or pulsed light, Proc. Phys. Soc., 83, 207-216. 

The Hanle effect and the theory of resonance fluorescence experiments... 

The theory of resonance fluorescence experiments therefore reduces to a study of the solutions of the Liouville equation for appropriate forms of the Hamiltonian operator 30. 

Introduction and experimental technique. The classical theory of resonance fluorescence, in which the atoms are treated as dipole oscillators processing at the Larmor frequency, leads one to predict that interesting effects will also occur if the atoms are excited by light whose intensity is periodically modulated. As the external magnetic field is varied in these experiments a point is reached at which the Larmor frequency, equals the angu-... 

Theory of resonance fluorescence excited by modulated light. Theoretical expressions for the intensity of light observed in resonance fluorescence experiments using modulated excitation can be obtained by a simple extension of equation (15.41). We now assume that the energy density of the incident radiation, U((o,t), is amplitude modulated at... 

The theory of resonance transfer of electronic excitation energy between donor and acceptor molecules of suitable spectroscopic properties was first presented by Forster.(7) According to this theory, the rate constant for singlet energy transfer from an excited donor to a chromophore acceptor which may or may not be fluorescent is proportional to r 6, where r is the distance... 

They found that Forster s theory of resonance energy transfer was applicable to such systems if the Interchromophoric distance and orientation, the fluorescence efficiency of the donor, the extinction of the acceptor, and the overlap between the emission of the donor and the absorption of the acceptor are of magnitudes which produce a transfer rate constant of less than 10 s and a transfer efficiency which is not too close to 0 or 1. 

The calculation of fluorescence yields for AFS are similar to those for molecular fluorescence (Chapter 5). Ingle and Crouch present an extensive discussion of theory of atomic fluorescence. Given the limited commercial applications of AFS, the theory will not be covered here. It is sufficient to understand that for a resonance transition and low analyte concentration, the fluorescence signal is proportional to the analyte concentration and to the intensity of the source. This assumption is valid for sources that do not alter the population of the analyte states. Intense laser sources can deplete the population of lower-energy states, including the state from which excitation occurs. This condition is called saturation and is discussed under applications of AFS in Section 7.6.3. 

This chapter opens with an account of resonance fluorescence and its depolarization by external magnetic fields, a phenomenon now knovm as the Hanle effect. Experiments of this type in mercury vapour are described and we develop a classical theory to explain the shape of the observed signals. This is followed by a discussion of the applications of this technique to the accurate measurement of atomic lifetimes. For the sake of simplicity the effects of interatomic collisions and of trapping or reabsorption of resonance radiation in these experiments are not considered... 

Next we proceed to develop the theory o resonance fluorescence experiments using the ensemble density matrix to describe the system of atoms. The important concepts of optical and radio-frequency coherence and of the interference of atomic states are discussed in detail. As an illustration of this theory general expressions describing the Hanle effect experiments are obtained. These are evaluated in detail for the frequently employed example of atoms whose angular momentum quantum numbers in the ground and excited levels are J =0 and Jg=l respectively. Finally resonance fluorescence experiments using pulsed or modulated excitation are described. 

Introduction and experimental techniques. In previous sections we drew attention to the fact that, in both the classical and quantum theories, expressions derived for the intensity of resonance fluorescence from atoms subjected to an external magnetic field, equations (15,3) and (15.23) respectively, contain terms which may lead to a modulation of the intensity at the Larmor frequency or its second harmonic. This radio-frequency modulation has been observed in several different kinds of experiment, the simplest of which makes use of pulsed excitation and time-resolved detection of the fluorescent light. 

F. Perrin Discussion on Jean Perriris diagram for the explanation of the delayed fluorescence by the intermediate passage through a metastable state First qualitative theory of fluorescence depolarization by resonance energy transfer... 

Resonance coupling, Fe—S dimers, 38 446-451 Resonance delocalization, 38 426 Resonance effects, of astatophenols, 31 66 Resonance fluorescence, 46 156 nuclear, 6 438-445 theory of, 6 433 38... 

This chapter is intended to provide an overview of the results obtained from recent experimental studies of sensitized fluorescence and quenching of resonance radiation in alkali and mercury vapors, up to 1974. The many inconsistencies and discrepancies that still exist between experiment and theory as well as among the experimental results themselves, and that are a source of concern to those working in the field, will no doubt continue to be resolved as new ideas are put forward and as both theoretical and experimental techniques advance in precision and sophistication. 

---