Introduction We have now prepared the way for a quantum treatment of problems that are of interest in optics. We have examined some basic properties of the quantized field and have explored the coherent-state representation and some of its virtues. Until now our discussions have been essentially formal; we have not concerned ourselves with what is actually measured in the laboratory, nor with questions concerning the interactions of the quantized field with other systems. In the following sections we shall turn our attention to some of the quantum field theoretic quantities that are of particular interest in quantum optics and are of significance in the measurement of the field. Our discussions in this chapter will still be somewhat formal, in that we shall not yet come to grips with the problem of the interaction of the field with the apparatus. That problem will be tackled in Chapter 14. For the moment we shall continue to oversimplify and treat the field as a free field. However, within the idealized context of the free field we shall examine some of the questions encountered in measurements in quantum optics. We shall see that normally ordered correlations play a dominant role and examine some of their properties. Anti-normally ordered correlations will be seen to arise in much less common and less useful measurements of the field. We shall also examine the sense in which a photon may be said to be localized in a measurement in which a photoelectric emission is registered.
Introduction We have now prepared the way for a quantum treatment of problems that are of interest in optics. We have examined some basic properties of the quantized field and have explored the coherent-state representation and some of its virtues. Until now our discussions have been essentially formal; we have not concerned ourselves with what is actually measured in the laboratory, nor with questions concerning the interactions of the quantized field with other systems. In the following sections we shall turn our attention to some of the quantum field theoretic quantities that are of particular interest in quantum optics and are of significance in the measurement of the field. Our discussions in this chapter will still be somewhat formal, in that we shall not yet come to grips with the problem of the interaction of the field with the apparatus. That problem will be tackled in Chapter 14. For the moment we shall continue to oversimplify and treat the field as a free field. However, within the idealized context of the free field we shall examine some of the questions encountered in measurements in quantum optics. We shall see that normally ordered correlations play a dominant role and examine some of their properties. Anti-normally ordered correlations will be seen to arise in much less common and less useful measurements of the field. We shall also examine the sense in which a photon may be said to be localized in a measurement in which a photoelectric emission is registered.
Interactions of a quantized electromagnetic field In the preceding chapters we have studied the quantum properties of the electromagnetic field, but we have treated it as a free or non-interacting quantum system until now. However, both the emission and the absorption of light imply interactions with other quantum systems, and we now turn to the treatment of such interaction problems. It is true that, to a limited extent, we have already succeeded in treating some interactions of light in previous chapters, without invoking the full apparatus of the quantum theory of interacting systems. For example, in Chapter 9 the electromagnetic field was treated as a classical potential acting on an atomic quantum system, and in Sections 12.2 and 12.9 we invoked simple heuristic arguments to describe the operation of certain optical detectors. But these are limited applications, and, in any case, the validity of results obtained in this way needs to be confirmed. To describe the state of a quantized electromagnetic field (F) in interaction with some other quantum system (A), we evidently require an enlarged, or product Hilbert space, which encompasses both F and A, of which the Hilbert spaces of F and A are subspaces. We shall find it convenient to refer to the other system A as an ‘atomic system’, simply to give a name, without restricting the nature of A. The dynamical variables of the field Ô F( t ) and of the atomic system Ô A( t ) commute at the same time t , and at the beginning, when the interaction between them is assumed to be turned on, each operator acts only on the state vectors within its respective Hilbert space.
Introduction In the preceding chapters we have been largely concerned with the simplest coherence effects of optical fields, namely those which depend on the correlation of the field variable at two space-time points ( r 1 , t 1 ) and ( r 2 , t 2 ). As we have seen, these effects include the most elementary coherence phenomena involving interference, diffraction and radiation from fluctuating sources. In this chapter we present an extension of the theory to cover more complicated situations, which have to be described by correlations of higher order, i.e. by correlations of the field variables at more than two space-time points or the expectation values involving various powers and products of the field variables. Situations of this kind have become of considerable importance since the development of the laser and of nonlinear optics. The basic difference between the statistical properties of thermal light and laser light can, in fact, only be understood by going beyond the elementary second-order correlation theory. Many of the higher-order coherence phenomena are most clearly manifest in the photoelectric detection process, which can only be adequately described by the quantum theory of detection or by taking into account the quantum features of the field, both of which will be studied in the succeeding chapters. However, because the field is still described classically in the semi-classical theory of light detection, and also because the classical field description provides a natural stepping stone to the quantum description of field correlations, we will now discuss the general description of field correlations of all orders on the basis of the classical theory of the fluctuating wavefield.
The properties of a photon counter functioning by stimulated emission---rather than by absorption---of photons in an external field are examined. A possible scheme for such a quantum counter is described, and it is shown that correlation measurements performed with a number of quantum counters correspond to antinormally ordered products of field operators. These correlations, unlike the normally ordered ones, are always positive definite and depend explicitly on the number of radiation modes per unit volume to which the counter is coupled. It is shown that the antinormally ordered correlations carry useful information about the field only when the average photon occupation number per mode is large. A general expression for the probability that the quantum counter registers n counts in a certain time interval is derived and is shown to be related in an interesting way to the corresponding expression for the photoelectric detector. The variance of the probability distribution is evaluated for some simple states of the field.
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.
Conceptual problems in quantum optics are investigated by a study of the physical significance of various types of description of the field of a single mode. The descriptions are examined through the effect of the field on matter. Both undamped and damped modes are considered, the descriptions being those of various degrees of precision in classical and quantum-mechanical formalisms. The matter consists of a large number of "molecules," and the interaction is studied by means of low-order perturbation theory. Formal expressions are obtained for the energy interchange between field and molecules in terms of the zeroth-order (uncoupled) operators, with the effect of the commutation properties of the field variables explicitly separated from that of the excitation of the field, in second order. The significance of taking expectation values of the operator results is discussed, and it is concluded that this procedure is justified only when the experiment under consideration involves an averaging process; thus, the molecular variables should be averaged, but not necessarily those of the field. Conditions under which the averaging process is justified are considered, and it is reasoned that, when such a process is not justified, a statistical description of the field can result only in a statistical description of the interaction process. Results of various field descriptions are compared, and it is shown that for induced emission and for spontaneous emission from coherently oscillating molecules, classical and quantum-mechanical fields are equivalent. However, for a self-consistent theory of spontaneous emission from molecules that are not oscillating coherently, the quantum-mechanical commutation properties of the field are essential if the molecules are treated quantum-mecahnically, and the field must be treated classically if the molecules are treated classically; the type of description used for the excitation of the field is inconsequential. Photoelectric emission is examined, and, within the approximation framework used, shown to be accurately described by a classical treatment of the field. It is pointed out that there does not necessarily exist a connection between "photon statistics," that is, the statistics associated with an ensemble of fields, and photocurrent fluctuations in one or more detectors coupled to the same field.
Statistical and quantum optics provides a clear, up-to-date and comprehensive introduction to the recent field of modern optics dealing with coherence and fluctuations of light. The first two chapters discuss the statistical frame work needed for later part of the book followed by systematic account of coherence theory with a detailed discussed on photon statistics. Description of quantum theory of coherence is preceded by a introductory chapter on field quantization. A detailed description of non-classical light highlighted by phenomena of anti-bunching, Sub-Poissonian statistics, production and detection of squeezed light and coherent state follows. Next is the discussion on E-P-R paradox on reality, locality and completeness of quantum mechanics bringing out clearly the importance of Bell's inequalities. The concluding two chapters put forward forcefully the new fundamental concepts based on quantum superposition and entanglement leading to two photon interference, quantum cryptography and teleportation.