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Physical Review A

Implications of radiative equilibrium in neoclassical theory

Franklin R. Nash, J. P. Gordon

Published Dec 1, 1975• 6 citations
Publisher

Abstract

We discuss some implications of the description of spontaneous emission of radiation which has been offered by Jaynes and his collaborators in their "neoclassical" extension of semiclassical electrodynamics. In particular, we examine the thermal-equilibrium condition of radiation interacting with a tenuous gas of atoms. We argue that rate equations may be used to describe the interaction of such atoms with the chaotic thermal radiation field. For this situation the neoclassical spontaneous emission rate is incompatible with the well-secured laws of Boltzmann and Planck. Experimental evidence bearing on the accuracy of those laws as well as on the accepted level population dependence of the induced emission rate is reviewed.

DOI
10.1103/physreva.12.2472

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We discuss some implications of the description of spontaneous emission of radiation which has been offered by Jaynes and his collaborators in their "neoclassical" extension of semiclassical electrodynamics. In particular, we examine the thermal-equilibrium condition of radiation interacting with a tenuous gas of atoms. We argue that rate equations may be used to describe the interaction of such atoms with the chaotic thermal radiation field. For this situation the neoclassical spontaneous emission rate is incompatible with the well-secured laws of Boltzmann and Planck. Experimental evidence bearing on the accuracy of those laws as well as on the accepted level population dependence of the induced emission rate is reviewed.
It has recently been shown by Jaynes and collaborators that semiclassical radiation theory contains a description of spontaneous emission of radiation and of radiative level shifts. The present paper gives a gauge-invariant derivation of the radiant energy production rate, using only Maxwell's equations and the usual definitions of electric current and charge density for a many-particle material system described by Schr\"odinger's equation. A complementary derivation, using the time-dependent Schr\"odinger equation, verifies the instantaneous conservation of energy and probability during the radiation process. The semiclassical rate of spontaneous emission differs from the usual formula (Einstein's AAA coefficient) because it depends on the occupancy of both initial and final states of the material system. The implications of this with regard to thermal equilibrium and Planck's law are examined. If a new hypothesis is introduced, postulating the decomposition of the equilibrium radiation into incoherent components, each interacting with a specific pair of energy levels of the material system, then Planck's law is shown to hold for the total intensity of radiation. If this hypothesis is not introduced, the equilibrium conditions for different transitions are incompatible, and the semiclassical radiation theory is incapable of describing thermal equilibrium.
The neoclassical theory of the interaction of radiation with matter, introduced by its authors as a potentially viable alternative to quantum electrodynamics, is examined in the case of a multilevel atom interacting with blackbody radiation. The stationary statistical states of the atom and the field are derived, and two independently untenable results emerge. First, the stationary blackbody law turns out to be that of Rayleigh and Jeans, rather than that of Planck. Second, the mean energy of a harmonic oscillator, which interacts with the field only near a single frequency, turns out to be infinite. These results are discussed in the light of the neoclassical description of spontaneous emission.
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 39K^{39}\mathrm{K}39K the decay rate is in agreement with the predictions of quantum electrodynamics and definitely in disagreement with the neoclassical theory.
When you follow two separate chains of thought, Watson, you will find some point of intersection which should approximate the truth. —Sir Arthur Conan Doyle, The Disappearance of Lady Frances Carfax ABSTRACT. It is now well established that effects traditionally associated with vacuum electromagnetic field fluctuations can be described equally well in terms of source fields (radiation reaction). This remarkable reconciliation of two previously unconnected points of view, which was stimulated by Jaynes' neoclassical theory, is reviewed and explained in a general and simple way. Background The Jaynes-Cummings paper seems to have been the first to employ dressed states of two-state atoms in fields (Jaynes 1963). One result found in that paper is that semiclassical radiation theory could serve as an excellent approximation even under certain conditions where the average number of photons in the field is small. The accuracy of semiclassical theory paved the way to Jaynes' “neoclassical theory,” where even spontaneous emission was treated without field quantization (Jaynes 1972, Milonni 1976). The neoclassical theory was a subject of much debate and misunderstanding in the 1970s. It turned out not to be so easy to dismiss it on experimental grounds.
Introduction The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E 2 − E 1 = ħ ω 21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters. Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results. Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħ ω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission. Einstein A - and B -coefficients Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied Δ E = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.
The Jaynes-Cummings paper treats the problem of one-atom interacting with one cavity mode with two alternative theories: quantum electrodynamics and semiclassical theory, and compares the predictions. The semiclassical theory was later extended to treat a single atom in free space interacting with a classical field including radiation reaction. The resulting theory was called Neoclassical theory, and is described by nonlinear optical Bloch equations. These nonlinear equations have shown up in a number of physical systems and were the subject of the famous Jaynes-Franken bet that was a prominent feature of the early Rochester Coherence and Quantum Optics Conferences. We will touch on all of these topics in a historical review.