In dealing with electromagnetic phenomena and in particular the phenomena of optics, despite the recognition of the quanta of light people tend to talk of the amplitudes and field strengths, as if the electromagnetic field were a classical field. For example we measure the wavelength of light by studying interference fringes. In this paper we study the inter-relationship of three ways of looking at the problem: in terms of classical wave fields, wave function of the photon; and the quantized wave field. The comparison and contrasts of these three modes of description are carefully analyzed in this paper. The ways in which these different modes complement our intuition and insight are also discussed.
In dealing with electromagnetic phenomena and in particular the phenomena of optics, despite the recognition of the quanta of light people tend to talk of the amplitudes and field strengths, as if the electromagnetic field were a classical field. For example we measure the wavelength of light by studying interference fringes. In this paper we study the inter-relationship of three ways of looking at the problem: in terms of classical wave fields, wave function of the photon; and the quantized wave field. The comparison and contrasts of these three modes of description are carefully analyzed in this paper. The ways in which these different modes complement our intuition and insight are also discussed.
In dealing with electromagnetic phenomena and in particular the phenomena of optics, despite the recognition of the quanta of light people tend to talk of the amplitudes and field strengths, as if the electromagnetic field were a classical field. For example we measure the wavelength of light by studying interference fringes. In this paper we study the inter- relationship of three ways of looking at the problem: in terms of classical wave fields, wave function of the photon; and the quantized wave field. The comparison and contrasts of these three modes of description are carefully analyzed in this paper. The ways in which these different modes complement our intuition and insight are also discussed.
Introduction Up to now the electromagnetic field has been treated as a classical field, describable by c -number functions. The great success of classical electromagnetic theory in accounting for a variety of optical phenomena, particularly those connected with wave propagation, interference and diffraction, amply justifies the classical approach. Moreover, as we have seen in the preceding chapters, in some cases the classical wave theory also gives a good account of itself in the treatment of the interaction of electromagnetic fields. For example, it is able to describe such seemingly non-classical effects as photoelectric bunching and the photo-electric counting statistics. It might almost seem that there is little justification for going beyond the domain of classical wave theory in optics. On the other hand, it can be argued that optics lies well and truly in the quantum domain, in the sense that we often encounter situations in which very few quanta or photons are present. In the microwave region of the electromagnetic spectrum, and at still longer wavelengths, the number of photons in each mode of the field is usually very large, and we are justified in treating the system classically. However, in the optical region the situation is usually just the opposite. As we show in Section 13.1, for light produced by practically all sources other than lasers, the average number of photons per mode is typically much less than unity.
In this chapter we shall be using the formalism of Chapter 4 to describe some properties of the quantized free electromagnetic field, emphasizing the analogies and differences with classical electromagnetism. The analogies explain why classical optics was so successful, to the extent that, even though light was not quantized, it was able to describe almost all known optical phenomena up until the 1970s. The differences highlight typical quantum behaviour, with no classical equivalent, which lies at the heart of modern quantum optics. The chapter begins by presenting formal ways of describing the main components of quantum optics experiments, namely, photodetectors and semi-reflecting mirrors. One can then understand homodyne detection, used to measure field quadrature components, even at optical frequencies for which no detector is fast enough to follow the oscillations of the field itself. Section 5.2 deals with the quantum vacuum where, in contrast to the classical vacuum, radiation has properties, in particular, fluctuations, with which one can associate physical effects. We then discuss in Section 5.3 some quantum radiation states in the simple case where a single mode l of the field is excited (non-vacuum). We introduce the idea of a quasiclassical state, or coherent state, used to make the link with classical optics. We shall also study other states, those with a definite number of photons, or squeezed states, which exhibit typically quantum properties, quite inconceivable in classical electromagnetism, and we shall show how these properties can be understood by reference to the Heisenberg relations, with an extremely useful graphical representation.
Quantum mechanical uncertainty After reviewing the main characteristics of the classical description of light, let us discuss those aspects of the quantization of the electromagnetic field which are of relevance for the analysis of the phenomena we are interested in. It seems a reasonable place to start to make clear the fundamental difference between the classical and the quantum mechanical description of nature; we will come across this difference many times when discussing experiments, and it will often give us a headache. We have to deal with the physical meaning of what is called uncertainty. The starting point of the classical description is the conviction that natural processes have a “factual” character. This means that physical variables such as the position or momentum of a particle have, in each single case, a well defined (in general, time dependent) value. However, it will not always be possible to measure all the appropriate variables (for instance, the instantaneous electric field strength of a radiation field); furthermore under normal circumstances we are able to measure only with a finite precision. Hence the basic credo of classical physics should be given in the following form: we are justified in imagining a world with variables possessing well defined values which are not known precisely (or not known at all). In doing this we are not forming any conclusions that contradict our everyday experiences. This is the fundamental concept of classical statistics: we are satisfied with the use of probability distributions for the variables we are interested in, not from fundamental but purely practical reasons.
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).
The present work is concerned with de Broglie's quantum theory of light. It is assumed that the photon is described by a Hermitian wave function with 16 components. Using this wave function it is shown that the 32 of de Broglie's equations are reduced to one set of 16 equations in the form: [equation] where the [alpha]'[rho] are Dirac matrices and [psi] is a matrix with 16 components. The electromagnetic quantities associated with the photon are described by means of the Dirac matrices operating on [psi] in a specified way. It is shown also that these electromagnetic quantities satisfy Maxwell's equations as a result of the equation the interaction between an electron and a photon is developed and the matrix elements for the radiation transitions are calculated. It is further shown that the above wave equation can be considered as the superposition of two similar wave equations, one for the positive energy photons and the other for the negative energy photons. To each of these states there corresponds electromagnetic quantities defined by the above method. It is the superposition of these fields which gives rise to the reality of the electromagnetic field, found in experience. The wave mechanics of the positive energy photon is discussed and the method of second quantization is applied to its wave function, from which we deduce the commutation relations for the complex fields.