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Molecular Physics

A perturbative formalism for the spontaneous emission by excited systems

Paolo Grigolini A

Published Jun 1, 1976• 30 citations
Publisher

Abstract

In this paper we improve a theory that we have previously developed in order to deal with the spontaneous decay of excited systems. These improvements allows us to eliminate some unpleasant features of the previous approach as well as to study the decay of excited systems in the presence of external irradiation fields and the response of matter to electromagnetic excitations. The present approach allows for a quantitative evaluation of the irradiation field influence on the state lifetimes, which (when the interaction under study is endowed with a ‘strong memory’) is shown to be noticeable even in the cases where the irradiation intensity is small with respect to the unperturbed linewidth. We expect that, taking into account also the third-order perturbation contributions, here disregarded, the slow-motion lineshape problem will be effectively treated by the present approach even in the saturation regime.

DOI
10.1080/00268977600101381

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In this paper we improve a theory that we have previously developed in order to deal with the spontaneous decay of excited systems. These improvements allows us to eliminate some unpleasant features of the previous approach as well as to study the decay of excited systems in the presence of external irradiation fields and the response of matter to electromagnetic excitations. The present approach allows for a quantitative evaluation of the irradiation field influence on the state lifetimes, which (when the interaction under study is endowed with a ‘strong memory’) is shown to be noticeable even in the cases where the irradiation intensity is small with respect to the unperturbed linewidth. We expect that, taking into account also the third-order perturbation contributions, here disregarded, the slow-motion lineshape problem will be effectively treated by the present approach even in the saturation regime.
A two-level atom interacting simultaneously with an external static field is reinvestigated. Here, first the Hamiltonian of the atom and the external field is diagonalized and then the interaction of the radiation field is considered. This method yields, directly, the state populations for the atom, and the lineshape and linewidth of the spontaneously emitted radiation. The earlier result on the state populations by Bali and Higgins has been compared with the present work and found to be in good agreement. The line shape of the emitted radiation is Lorenzian and the halfwidth is found to be increasing with increasing external perturbation. For the case where the atom is initially in the excited state, the intensity of the emitted radiation decreases with increasing perturbation and finally reaches a saturation limit. However, when the atom is considered to be in the ground state, initially, the intensity increases with increasing perturbation and it again reaches a saturation limit at a large perturbation field.
The spontaneous radiative decay to the ground state of an atomic system initially in the higher of two excited states, coupled in a radiationless fashion by an external field, is investigated by the method of multiple-time-scale perturbation theory. The coupled differential equations of motion for the probability amplitudes are solved to second order, while the usual secular behavior of conventional time-dependent perturbation theory is eliminated in terms of two natural time scales: a fast time scale corresponding to the reciprocal of the level transition frequency and a slower time scale associated with the inverse of the radiation linewidth. Expressions are obtained to this order for the energy shifts and decay characteristics of the excited states and the energy distribution of the final-state photons, quantities previously determined by Fourier analysis and contour integration and by the phenomenological approach of Weisskopf and Wigner. A detailed comparison of these various methods of solution includes conditions under which the present results agree to second order with the previous calculations.
The problem of the spontaneous emission from a single excited two-level atom in the presence of N unexcited similar atoms is discussed. A formalism is presented for evaluation of the time-development operator (and thus the density matrix) which has certain desirable features for time-dependent problems in the area, e.g. of atom-field interactions, when the off-diagonal elements are large compared to the diagonal elements of the Hamiltonian. By use of this formalism, it is shown that the spontaneous emission from a single atom is often drastically altered in the presence of unexcited atoms with similar resonances. When the number of atoms present becomes very large relative to the number of modes of the electromagnetic field, the excited atom becomes trapped in its excited state.>
A theory describing the radiative decay of an autoionizing state under strong pumping by a coherent field is developed. The theory systematically takes into account the radiative decay of the unperturbed continuum. The problem at hand corresponds to the case of strongly coupled bound states decaying to electron and photon continua with the two continua also weakly coupled to each other. A master equation describing the time evolution of the atomic system is derived, and its exact solution under arbitrary initial conditions is given. The effect of radiative decay on the Fano profiles and photoelectron spectra is analyzed in detail. The time development of the system is also examined. The radiative decay of the autoionizing state and the unperturbed continuum changes the spectra in a significant way. The characteristics of the spectra are correlated with the dressed states (with complex energies) of the system. The changes in the structure of the dressed states as a function of the system parameters such as the spontaneous-emission rate and laser intensity are discussed in detail.
So far we have treated the energy levels of an atomic system as stable states with infinite lifetimes. The presence of an electromagnetic field changes this picture. Absorption of photons will excite atoms and ions to higher levels, while stimulated emission will make excited ones decay. Einstein (1917) showed, using purely statistical arguments, that an excited ion will decay through spontaneous emission in the absence of an electromagnetic field. This process can also be derived by using a rigorous quantum electrodynamic treatment, where the field is quantized. In this chapter we start by discussing transitions between atomic levels in a general sense, postulating the operators responsible for the decay. After the first few sections we will concentrate on the most important few multipoles or terms in the expansion of the electromagnetic field. We will introduce a number of different properties describing the transition, and eventually arrive at the measurable transition rate and, directly linked to it, the lifetime of an excited state. We will start, however, with the most basic property, the component strength. Our aim is to arrive at computationally useful expressions as soon as possible. For a more thorough treatment of the theory of radiative transitions we refer to the book by Sobelman (1972). For a brief review of the many factors affecting spectra of atoms, see Martin and Wiese (1996) .