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arXiv:physics/0203051

The zero point field and the reduction of the wave packet in semiclassical electrodynamics

Jacques Moret‐Bailly

Published Mar 18, 2002• 0 citations
arXivHTMLPDF

Abstract

In the classical theory, an electromagnetic field obeying Maxwell's equations cannot be absorbed quickly by matter, so that it remains a zero point field. Splitting the total, genuine electromagnetic field into the sum of a conventional field and a zero point field is physically meaningless until a receiver attenuates the genuine field down to the zero point field, or studying the amplification of the zero point field by a source. In classical optics all optical effects must be written using the genuine field, so that at low light levels the nonlinear effects become linear in relation to the conventional field. The result of the interpretation of all observations, even at low light levels, is exactly the same in quantum electrodynamics and in the semi- classical theory. The zero point field is stochastic only far from the sources and the receivers; elsewhere, it is shaped by matter, it may be studied through fields visible before an absorption or after an amplification. A classical study of the reduction of the wave packet extends the domain of equivalence of the classical and quantum zero point field; using both interpretations of this field makes the results more reliable, because the traps are different.

DOI
10.48550/arxiv.physics/0203051

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In the classical theory, an electromagnetic field obeying Maxwell's equations cannot be absorbed quickly by matter, so that it remains a zero point field. Splitting the total, genuine electromagnetic field into the sum of a conventional field and a zero point field is physically meaningless until a receiver attenuates the genuine field down to the zero point field, or studying the amplification of the zero point field by a source. In classical optics all optical effects must be written using the genuine field, so that at low light levels the nonlinear effects become linear in relation to the conventional field. The result of the interpretation of all observations, even at low light levels, is exactly the same in quantum electrodynamics and in the semi- classical theory. The zero point field is stochastic only far from the sources and the receivers; elsewhere, it is shaped by matter, it may be studied through fields visible before an absorption or after an amplification. A classical study of the reduction of the wave packet extends the domain of equivalence of the classical and quantum zero point field; using both interpretations of this field makes the results more reliable, because the traps are different.
The zero point field is an ordinary field existing in the dark, which cannot be separated from the total electromagnetic field in an excited mode. The total field is in equilibrium with matter that it polarizes temporarily and reversibly. This polarisation is exceptionally large enough to allow the energy of an atom reach and cross a pass between two minimums of potential, stimulating an emission or an absorption. A paradox of quantum electrodynamics is explained classically: The nearly plane wave generally used in stimulated emission experiments must be decomposed into an efficient spherical wave and a scattered wave, so that the plane wave is two times less efficient than the zero point component of the spherical mode. The classical electromagnetism does not need the quantum postulate "reduction of the wave packet" and it has no paradox such as EPR. De Broglie's waves may be the linear part of solitons of a field which may be a high frequency electromagnetic field. These solitons may lead to a physical interpretation of the superstrings theory.
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’).
How can we construct a picture of the photon from the wealth of observation material available to us? The photon appears to have a split personality: it is neither a wave nor a particle but something else which, depending on the experimental situation, exhibits a wave- or a particle-like behavior. In other words, in the photon (as in material particles such as the electron) the particle–wave dualism becomes manifest. Whereas classically the wave and the particle pictures are separate, quantum mechanics accomplishes a formal synthesis through a unified mathematical treatment. Let us look first at the wave aspect familiar from classical electrodynamics, which seems to be the most natural description. It makes all the different interference phenomena understandable, such as the “interference of the photon with itself” on the one hand and the appearance of spatial and temporal intensity correlations in a thermal radiation field on the other (which are obviously brought about by superposition of elementary waves emitted independently from different atoms). It might come as a surprise (at least for those having quantum mechanical preconceptions) that the classical theory is valid down to arbitrarily small intensities: the visibility of the interference pattern does not deteriorate even for very small intensities – the zero point fluctuations of the electromagnetic field advocated by quantum mechanics do not have a disturbing effect – and is valid not only for conventional interference experiments but also for interference between independently generated light beams (in the form of laser light).
In this introductory chapter we focus on the interaction of optical fields with matter because it forms the basis of signal amplification in all optical amplifiers. According to our present understanding, optical fields are made of photons with properties precisely described by the laws of quantum field theory [1]. One consequence of this wave–particle duality is that optical fields can be described, in certain cases, as electromagnetic waves using Maxwell's equations and, in other cases, as a stream of massless particles (photons) such that each photon contains an energy h ν, where h is the Planck constant and ν, is the frequency of the optical field. In the case of monochromatic light, it is easy to relate the number of photons contained in an electromagnetic field to its associated energy density. However, this becomes difficult for optical fields that have broad spectral features, unless full statistical features of the signal are known [2]. Fortunately, in most cases that we deal with, such a detailed knowledge of photon statistics is not necessary or even required [3, 4]. Both the linear and the nonlinear optical studies carried out during the last century have shown us convincingly that a theoretical understanding of experimental observations can be gained just by using wave features of the optical fields if they are intense enough to contain more than a few photons [5]. It is this semiclassical approach that we adopt in this book. In cases where such a description is not adequate, one could supplement the wave picture with a quantum description.
The classical free-space solutions of Maxwell’s equations for light propagation in one dimension include wave packets of any shape that travel at the speed of light. This includes highly-localised wave packets that remain localised at all times. Motivated by this observation, this paper builds on recent work by Southall et al. [J. Mod. Opt. 68, 647 (2021)] and shows that a local description of the quantised electromagnetic field, which supports such solutions and which must overcome several no-go theorems, is indeed possible. Starting from the assumption that the basic building blocks of photonic wave packets are so-called bosons localised in position (blips), we identify the relevant Schrödinger equation and construct Lorentz-covariant electric and magnetic field observables. In addition we show that our approach simplifies to the standard description of quantum electrodynamics when restricted to a subspace of states.