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Physical Review A

Contradictions with the Neoclassical Theory of Radiation in Weakly Excited Multilevel Systems

H. M. Gibbs, G. G. Churchill, G. J. Salamo

Published May 1, 1973• 15 citations

Abstract

The neoclassical or semiclassical theory (NCT) of the radiative decay of an isolated atom predicts that the decay rate of an excited state is proportional to the atoms's population in the state to which the atom is decaying. For weak excitation of a two-level system, this prediction is consistent with observations and with the Weisskopf-Wigner theory of decay, which assumes a quantized electromagnetic field as in quantum electrodynamics (QED). But for weak excitation of a system with three or more levels, there can be gross disagreements between NCT and QED. NCT predicts that the lifetime of an excited state depends on the nuclear spin which determines the number of ground states between which the atom's unity population density is divided. Also NCT predicts no fluorescence to a third level not initially populated, even though the transition is highly allowed. Observations in Pb vapor are inconsistent with NCT and consistent with QED. The oscillator strengths of the 2833-, 3640-, and 4058-A transitions were determined to be 0.167(20), 0.061(04), and 0.146(10), respectively, by measuring the ratios of oscillator strengths and using the lifetime measurements of Happer and Saloman.

DOI: 10.1103/physreva.7.1766

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The neoclassical or semiclassical theory (NCT) of the radiative decay of an isolated atom predicts that the decay rate of an excited state is proportional to the atoms's population in the state to which the atom is decaying. For weak excitation of a two-level system, this prediction is consistent with observations and with the Weisskopf-Wigner theory of decay, which assumes a quantized electromagnetic field as in quantum electrodynamics (QED). But for weak excitation of a system with three or more levels, there can be gross disagreements between NCT and QED. NCT predicts that the lifetime of an excited state depends on the nuclear spin which determines the number of ground states between which the atom's unity population density is divided. Also NCT predicts no fluorescence to a third level not initially populated, even though the transition is highly allowed. Observations in Pb vapor are inconsistent with NCT and consistent with QED. The oscillator strengths of the 2833-, 3640-, and 4058-A transitions were determined to be 0.167(20), 0.061(04), and 0.146(10), respectively, by measuring the ratios of oscillator strengths and using the lifetime measurements of Happer and Saloman.

The neoclassical theory of radiation predicts that the decay rate of an excited atomic state depends on the population density of the lower state. Experimental evidence is presented here which shows that in the case of

^{39}\mathrm{K}

the decay rate is in agreement with the predictions of quantum electrodynamics and definitely in disagreement with the neoclassical theory.

The resonance-fluorescence experiment described in the preceeding paper and found to be consistent with quantum electrodynamics is analyzed using the neoclassical theory (NCT) of radiation, i.e., the electromagnetic field is not quantized. We have not found a value of the dynamic shift for which the NCT computer simulation is consistent with the data. NCT predicts an increased radiative lifetime as appreciable excitation is achieved; the observed decay curves were simple exponentials with characteristic times independent of the fractional excitation. The observed oscillatory dependence of the integrated fluorescence upon input pulse area has not been reproduced by a NCT simulation either. Objections raised against this disproof of NCT are answered.

In Jaynes's semiclassical radiation theory, the decay of an excited atomic state depends on its initial population. For the radiative decay of the

2P

level in atomic hydrogen, produced by pulsed excitation from the

2S

level, the theory predicts a decay rate $\ensuremath{\Gamma}=(1\ensuremath{-}f){\ensuremath{\Gamma}}_{0}+\ensuremath{\beta}P$, where $\ensuremath{\beta}=f{\ensuremath{\Gamma}}_{0}$. Here

f

is the initial

2S

population, ${\ensuremath{\Gamma}}_{0}$ is the natural

2P

decay rate, and

P

is the

2P

population relative to

f

. For $f=0.086\ifmmode\pm\else\textpm\fi{}0.016$, and

P

values in the range 0.15 to 0.64, we have measured $\ensuremath{\beta}=\ensuremath{-}(2.8\ifmmode\pm\else\textpm\fi{}2.8)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. The corresponding value from the semiclassical theory is $\ensuremath{\beta}=+(53.8\ifmmode\pm\else\textpm\fi{}10.0)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$.

The theory of spontaneous decay is studied using both quantum electrodynamics (QED) and semiclassical theories of radiation. There are qualitative differences between the theories in the prediction of interference phenomena. In QED, systems which have been excited with pulsed laser light do not exhibit quantum interference effects associated with lower state splittings. On the other hand, semiclassical treatments of spontaneous decay do indicate the existence of interference effects not present in QED. In addition to this, differences are found between the predictions of fluorescence intensity in the presence of lower-state level crossings under continuous excitation.

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) .