Nonlinear optics is a fairly young science, having taken off with the advent of the laser in 1960. Nonlinear optics (NLO) deals with the interaction of electromagnetic waves and matter in the infrared, visible, and ultraviolet domains. The frontiers of NLO are somewhat blurred, but microwaves and γ-rays are clearly outside its domain. This book is entirely devoted to a study of NLO in a resonant cavity and when the quantum nature of the electromagnetic field is not of prime importance. More precisely, we study those aspects of cavity NLO in which fluctuations in the number of photons and atoms are not relevant. This particular area of optics is dominated by the Maxwell–Bloch equations, which constitute its paradigm in the sense of T. S. Kuhn. The status of the Maxwell–Bloch equations is quite peculiar. From a fundamental viewpoint, they describe the laws of evolution of the first moments of a density operator, which verifies the von Neumann equation. However, to account for the finite lifetime of the atoms and of the field in the necessarily lossy cavity, some legerdemains have to be introduced to obtain the Maxwell–Bloch equations. Stated more explicitly, the von Neumann equation for a large but finite system does not explain irreversibility, whereas the Maxwell–Bloch equations fully include the irreversible decay of the atoms and of the cavity field. This problem is not specifically related to optics but reflects the general failure of statistical mechanics to explain convincingly the irreversible evolution of macroscopic systems.
Nonlinear optics is a fairly young science, having taken off with the advent of the laser in 1960. Nonlinear optics (NLO) deals with the interaction of electromagnetic waves and matter in the infrared, visible, and ultraviolet domains. The frontiers of NLO are somewhat blurred, but microwaves and γ-rays are clearly outside its domain. This book is entirely devoted to a study of NLO in a resonant cavity and when the quantum nature of the electromagnetic field is not of prime importance. More precisely, we study those aspects of cavity NLO in which fluctuations in the number of photons and atoms are not relevant. This particular area of optics is dominated by the Maxwell–Bloch equations, which constitute its paradigm in the sense of T. S. Kuhn. The status of the Maxwell–Bloch equations is quite peculiar. From a fundamental viewpoint, they describe the laws of evolution of the first moments of a density operator, which verifies the von Neumann equation. However, to account for the finite lifetime of the atoms and of the field in the necessarily lossy cavity, some legerdemains have to be introduced to obtain the Maxwell–Bloch equations. Stated more explicitly, the von Neumann equation for a large but finite system does not explain irreversibility, whereas the Maxwell–Bloch equations fully include the irreversible decay of the atoms and of the cavity field. This problem is not specifically related to optics but reflects the general failure of statistical mechanics to explain convincingly the irreversible evolution of macroscopic systems.
A rigorous description of the light–matter interaction requires the use of quantized Maxwell equations to describe the light field and the Schrödinger equation to describe the material medium. This book does not attempt to deal with quantum optical problems that necessitate a field quantization. Instead we focus on properties that are generally grouped under the umbrella of nonlinear optics . Although many definitions of this expression exist, we use it in the sense that the properties we deal with do not depend in a crucial way on the field quantization. For practical purposes, this means in general that the average photon number is large and that we can neglect spontaneous emission as a dynamical process. Rather, we include the consequence of spontaneous emission, that is, the instability of the atomic levels, in a purely phenomenological way. When this is done, we call the material equations the Bloch equations instead of the Schrödinger equation. References [1] through [4] are the classical textbooks that contain a tentative justification for the transition from the Schrödinger to the Bloch equations. None of these attempts is satisfactory from a fundamental viewpoint because the real difficulty is to incorporate the finite lifetime (or natural linewidth) of the atomic energy levels in the Schrödinger equation. This remains an open problem as of now. The Maxwell–Schrödinger equations We consider atoms interacting with an intense electromagnetic field. By intense we mean a field for which the quantization is not necessary.
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.
Abstract This book is an introduction to quantum optics for students who have studied electromagnetism and quantum mechanics at an advanced undergraduate or graduate level. It provides detailed expositions of theory with emphasis on general physical principles. Foundational topics in classical and quantum electrodynamics, including the semiclassical theory of atom-field interactions, the quantization of the electromagnetic field in dispersive and dissipative media, uncertainty relations, and spontaneous emission, are addressed in the first half of the book. The second half begins with a chapter on the Jaynes-Cummings model, dressed states, and some distinctly quantum-mechanical features of atom-field interactions, and includes discussion of entanglement, the no-cloning theorem, von Neumann’s proof concerning hidden variable theories, Bell’s theorem, and tests of Bell inequalities. The last two chapters focus on quantum fluctuations and fluctuation-dissipation relations, beginning with Brownian motion, the Fokker-Planck equation, and classical and quantum Langevin equations. Detailed calculations are presented for the laser linewidth, spontaneous emission noise, photon statistics of linear amplifiers and attenuators, and other phenomena. Van der Waals interactions, Casimir forces, the Lifshitz theory of molecular forces between macroscopic media, and the many-body theory of such forces based on dyadic Green functions are analyzed from the perspective of Langevin noise, vacuum field fluctuations, and zero-point energy. There are numerous historical sidelights throughout the book, and approximately seventy exercises.
For many contemporary physics experiments, the use of microwave and optical resonators has become a powerful tool for enhancement in detection sensitivities, nonlinear interactions, and quantum dynamics. Most often, the term cavity is used to describe such electromagnetic resonators. This term has been taken over from microwave technology, where resonators really look like closed cavities, whereas optical resonators traditionally have an open kind of setup. That difference in geometry is related to the fact that optical resonators are usually very large compared with the optical wavelength, whereas microwave cavities are often not much longer than a wavelength. As discussed in the course of this book, microwave and optical cavities allow one to extend the interaction length between matter and field, to build up the optical power, to impose a well-defined mode structure on the electromagnetic field, to implement extreme nonlinear optics, and to study manifestly quantum mechanical behavior associated with the modified vacuum structure and/or the large field associated with a single photon confined to a small volume [100].
The field of quantum optics has progressed rapidly in the last twenty five years with the advent of the laser. Over much of this period the phenomena studied could be described adequately by semiclassical treatments. Quite recently however, there has been a revival of interest in genuinely quantum mechanical effects. The Malvern Symposium of December 1985 brought together world experts for a meeting which concentrated largely on these quantum effects. The presentations in this unique meeting combine review material with the very latest results and so will be of value to students of quantum optics and measurement theory at all levels. The first articles cover the exciting topic of the generation of squeezed states of light in the laboratory, and their possible uses. Experimental success has been long sought and very recently attained. The reader will find presented the state of the art in this field. Next to lasing itself, optical bistability has been the most widely studied phenomenon in quantum optics, largely for its technological promise. However, it also provides a fundamental system to study quuantum effects. Recent theoretical studies of optical bistability with small numbers of atoms are surveyed. In such situations quantum features such as antibunching become significant, and the articles in this volume should be a guide to those venturing into this challenging area. In other articles discussions of fluctuations from other noise sources and instabilities in optical bistabilty are presented in a clear and interesting way. Perhaps the least classical state on quantum optics is that describing a single photon. Recent experiments which produce such states are reviewed. A theoretical review of the photon together with some new material is given which delves deeply into relativistic quantum field theory in order to describe the concept of weakly localised photon states. The material here is very rarely presented in the context of quantum optics. The history of the theory of the quantum fluctuations in a laser is then reviewed. An off-shoot of this theory is the study of quantum chaos in dissipative systems and recent results in this new area are given in a succeeding article. There are further stimulating articles on Rydberg atom systems and quantum electrodynamics. The volume ends with an entertaining and incisive study of quantum measurement problems, such as the Schrodinger cat papadox, using concepts and measuring devices found in quantum optics.other_titles
In this chapter, we remove one assumption that has been implicit since the beginning of this book, namely the unidimensional aspect of the cavity. We now take into account the transverse variation of the field in the resonant cavity. In dealing with transverse effects, there are two possible approaches, very much like in the multimode optical bistability (OB) studied in Chapter 6. One possibility is to project the Maxwell–Bloch equations on a suitable basis. In the introduction of Chapter 6, the difficulty of selecting this suitable basis was explained for 1-D cavities. This difficulty is amplified by the transverse dimensions, especially because of the lateral boundaries. The other approach is to derive global equations (that are still partial differential equations) for slowly varying amplitudes. They are generally variants of well-known nonlinear partial differential equations of mathematical physics, and therefore a large number of results are directly available. However, when transferring results from another domain to optics, some care must be exercised because the relevant domains of parameters are not always compatible. The classic example is the canonical set of parameters for the Lorenz equations that includes b = γ ║ /γ ⊥ = 8/3 whereas for atomic transitions the upper bound of b is 2. The reader will find a wealth of results, mostly for the modal expansions, in the special issues and reviews [1]–[4]. A review more specifically oriented toward the global amplitude equations is found in [5] and a mathematical study of these equations is presented in [6].