Quantum optics, the union of quantum field theory and physical optics, is undergoing a time of revolutionary change. The subject has evolved from early studies on the coherence properties of radiation like, for example, quantum statistical theories of the laser in the sixties to modern areas of study involving, for example, the role of squeezed states of the radiation field and atomic coherence in quenching quantum noise in interferometry and optical amplifiers. On the one hand, counter intuitive concepts such as lasing without inversion and single atom (micro) masers and lasers are now laboratory realities. Many of these techniques hold promise for new devices whose sensitivity goes well beyond the standard quantum limits. On the other hand, quantum optics provides a powerful new probe for addressing fundamental issues of quantum mechanics such as complementarity, hidden variables, and other aspects central to the foundations of quantum physics and philosophy.

The intent of this book is to present these and many other exciting developments in the field of quantum optics to students and scientists, with an emphasis on fundamental concepts and their applications, so as to enable the students to perform independent research in this field. The book (which has developed from our lectures on the subject at various universities, research institutes, and summer schools) may be used as a textbook for beginning graduate students with some background in standard quantum mechanics and electromagnetic theory. Each chapter is supplemented by problems and general references.

Optical interferometry was at the heart of the revolution which ushered in the new era of twentieth century physics. For example, the Michelson interferometer was used to show that there is no detectable motion relative to the ‘ether’; a key experiment in support of special relativity.

It is a wonderful tribute to Michelson that the same interferometer concept is central to the gravity-wave detectors which promise to provide new insights into general relativity and astrophysics in the twenty-first century. Similar tales can be told about the Sagnac and Mach–Zehnder interferometers as discussed in this chapter. We further note that the intensity correlation stellar interferometer of Hanbury- Brown and Twiss was a driving force in ushering in the modern era of quantum optics.

We are thus motivated to develop the theory of field (amplitude) and photon (intensity) correlation interferometry. In doing so we will find that the subject provides us with an exquisite probe of the micro and macrocosmos, i.e., quantum mechanics and general relativity.

With these thoughts in mind we here develop a framework to study the quantum statistical correlations of light. We will motivate the quantum correlation functions of the field operators from the standpoint of photodetection theory. Many experimentally observed quantities, such as photoelectron statistics and the spectral distribution of the field, can be related to the appropriate field correlation functions. These correlation functions are essential in the description of Young's double-slit experiment and the notion of the power spectrum of light.

In this chapter, we present a theory of the laser based on the Heisenberg–Langevin approach. This is a different, but completely equivalent approach to the density operator approach discussed in the previous chapter. In general, the density operator approach is better suited to study the photon statistics of the radiation field whereas the Heisenberg–Langevin approach has certain calculational advantages in the determination of phase diffusion coefficients, and consequently laser linewidth.

In Section 12.1, a simple approach to determine laser linewidth based on a linear theory is presented. This analysis is especially interesting and useful in that it includes atomic memory effects, something that is difficult to do within a density matrix theory. In Sections 12.2–12.4, we consider the complete nonlinear theory of the laser and rederive all the important quantities related to the quantum statistical properties of the radiation field.

A simple Langevin treatment of the laser linewidth including atomic memory effects†

The full nonlinear quantum theory of the laser discussed in the previous chapter yields most of the interesting quantum statistical properties of the radiation field. In many problems of interest, however, we do not need such an elaborate treatment. For example, as we saw in the previous chapter, the natural linewidth of the laser can be determined from a linearized theory of the laser. That is, the full nonlinear theory serves to determine the amplitude of the field but the phase fluctuations about this operating point are described by a linear theory.

One of the simplest nontrivial problems involving the atom–field interaction is the coupling of a two-level atom with a single mode of the electromagnetic field. A two-level atom description is valid if the two atomic levels involved are resonant or nearly resonant with the driving field, while all other levels are highly detuned. Under certain realistic approximations, it is possible to reduce this problem to a form which can be solved exactly; allowing essential features of the atom-field interaction to be extracted.

In this chapter we present a semiclassical theory of the interaction of a single two-level atom with a single mode of the field in which the atom is treated as a quantum two-level system and the field is treated classically. A fully quantum mechanical theory will be presented in Chapter 6.

A two-level atom is formally analogous to a spin-1/2 system with two possible states. In the dipole approximation, when the field wavelength is larger than the atomic size, the atom–field interaction problem is mathematically equivalent to a spin-1/2 particle interacting with a time-dependent magnetic field. Just as the spin-1/2 system undergoes the so-called Rabi oscillations between the spin-up and spin-down states under the action of an oscillating magnetic field, the two-level atom also undergoes optical Rabi oscillations under the action of the driving electromagnetic field. These oscillations are damped if the atomic levels decay. An understanding of this simple model of the atom–field interaction enables us to consider more complicated problems involving an ensemble of atoms interacting with the field.

In Chapters 5 and 7 we presented a treatment of laser physics in which the light is described as a classical Maxwell field while the lasing medium is described as a collection of atoms whose dynamic evolution is governed by the Schrödinger equation. This semi-classical theory of laser behavior is sufficient to describe a rich variety of phenomena. However, there are many questions which require a fully quantized theory of the radiation. For example, the photon statistics and linewidth of the laser can be properly understood only via the full quantum theory of a laser.

The laser linewidth is an important quantity. For example, it determines the fundamental limit of operation of an active ring laser gyroscope. The first fully quantized derivation of the laser linewidth general enough to include even the semiconductor laser linewidth problem utilized a quantum noise operator approach, and is presented in chapter 12.

The photon statistical distribution for the laser is of interest for several reasons. Historically, it was initially thought by some that the statistical photon distribution should be a Bose–Einstein distribution. A little reflection shows that this can not be, since the laser is operating far from thermodynamic equilibrium. However, a different paradigm recognizes many atoms oscillating in phase produce what is essentially a classical current, and this would generate a coherent state; the statistics of which is Possionian. But, for example, the photon statistics of a typical Helium–Neon laser is substantially different from a Possionian distribution.

As was demonstrated in the previous chapter, the process of observation and acquisition of information or at least the possibility of ‘knowing’ (whether or not we bother to ‘look’) can profoundly change the outcome of an experiment. For example, in the case of the micromaser which-path detector, we do not need to ‘look at’ or ‘interrogate’ the masers in order to lose the interference cross term; it is enough that we could have known. Experiments along these lines provide a dramatic example of the importance of which-path, or ‘Welcher-Weg’, information.

The present chapter treats the Welcher-Weg quantum eraser problem from a different vantage. We first consider the interference of light as it is scattered from simple atomic systems consisting of single atoms located at two neighboring sites. From this simple model, we can gain a wealth of insight into such problems as complementarity, delayed choice, and the quantum eraser via field–field and photon–photon correlation functions, i.e., via G(1)(r, t) and G(2)(r,r′t, t′). The chapter concludes with a demonstration that such considerations can, in principle, even lead to new kinds of high-resolution spectroscopy.

Quantum mechanics is an immensely successful theory, occupying a unique position in the history of science. It has solved mysteries ranging from macroscopic superconductivity to the microscopic theory of elementary particles and has provided deep insights into the nature of vacuum on the one hand and the description of the nucleon on the other. Whole new fields such as quantum optics and quantum electronics owe their very existence to this body of knowledge.

However, despite the stunning successes of quantum mechanics, there is no general agreement on the conceptual foundations and interpretation of the subject. The theory provides unambiguous information about the outcome of a measurement of a physical object. However, many feel that it does not provide a satisfactory answer to the nature of the “reality” we should attribute to the physical objects between the acts of measurement.

The conceptual difficulty comes about because the wave function |ψ〉 is usually given by a coherent superposition of various distinguishable experimental outcomes. If we denote the collection of states that represent the possible outcomes of an experiment by |ψj〉, then |ψ〉 = ∑jcj|ψj〉 where cj = 〈ψj|ψ〉. The probability of the outcome |ψj〉 is Pj=|cj|2. In the process of measurement, the so called collapse of the wave function takes place and a single, definite state |ψi〉 of the physical object is chosen. The difficulty comes about in the interpretation of the mechanism by which this definite state is chosen from amongst all the possible outcomes.

The development of a single-atom maser or a micromaser allows a detailed study of the atom–field interaction. The situation realized is very close to the ideal case of a single two-level atom interacting with a single-mode quantized field as treated in Section 6.2. In a micromaser a stream of two-level atoms is injected into a superconducting cavity with a high quality factor. The injection rate can be such that only one atom is present inside the resonator at any time. Due to the high quality factor of the cavity, the radiation decay time is much larger than the characteristic time of the atom–field interaction, which is given by the inverse of the single-photon Rabi frequency. Therefore, a field is built up inside the cavity when the mean time between the atoms injected into the cavity is shorter than the cavity decay time. A micromaser, therefore, allows sustained oscillations with less than one atom on the average in the cavity.

The realization of a single-atom maser or a micromaser has been made possible due to the enormous progress in the construction of superconducting cavities together with the laser preparation of highly excited atoms called Rydberg atoms. The quality factor of the superconducting cavities is high enough for periodic energy exchanges between atom and cavity field to be observed. The interesting properties of the Rydberg atoms make them ideal for micromasers. In Rydberg atoms the probability of induced transitions between adjacent states becomes very large and scales as n4, where n denotes the principle quantum number.

As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons.

First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution.

In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P-representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P-representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q-representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.

Atomic physics is one of the oldest fields of physics. A barren and “academic” discipline? Not at all! About ten years ago, atomic physics received a rejuvenating jolt from chaos theory with far reaching implications. Chaos in atomic physics is today one of the most active and prolific areas in atomic physics. This book, addressed at interested students and practitioners alike, is a first attempt to provide a coherent introduction into this fascinating area of contemporary research. In line with its scope, the book is essentially divided into two parts. The first part of the book (Chapters 1 – 5) deals with the theory and philosophy of classical chaos. The ideas and concepts developed here are then applied to actual atomic and molecular physics systems in the second part of the book (Chapters 6 – 10).

When compiling the material for the first part of the book we profited immensely from a number of excellent tutorials on classical and quantum chaos. We mention the books by Lichtenberg and Lieberman (1983), Zaslavsky (1985), Schuster (1988), Sagdeev et al. (1988), Tabor (1989), Gutzwiller (1990), Haake (1991), Devaney (1992) and Reichl (1992).

The illustrative examples for the second part of the book were mostly taken from our own research work on the manifestations of chaos in atomic and molecular physics. We apologize at this point to all the numerous researchers whose work is not represented in this book. This has nothing to do with the quality of their work and is due only to the fact that we had to make a selection.

The stability of the solar system is one of the most important unsettled questions of classical mechanics. Even a simplified version of the solar system, the three-body problem, presents a formidable challenge. An important breakthrough occurred when Poincaré, with some assistance from his Swedish colleague Fragmen, proved in 1892 that, apart from some notable exceptions, the three-body problem does not possess a complete set of integrals of the motion. Thus, in modern parlance, the three-body problem is chaotic.

The helium atom is an atomic physics example of a three-body problem. On the basis of Poincaré's result we have to expect that the helium atom is classically chaotic. Richter and Wintgen (1990b) showed that this is indeed the case: the helium atom exhibits a mixed phase space with intermingled regular and chaotic regions (see also Wintgen et al. (1993)). Thus, conceptually, the helium atom is a close relative of the double pendulum studied in Section 3.2. Given the classical chaoticity of the helium atom we are confronted with an important question: How does chaos manifest itself in the helium atom?

In order to provide clues for an answer to this question we study in this chapter a one-dimensional version of the helium atom, the “stretched helium atom” (Watanabe (1987), Blümel and Reinhardt (1992)). This model, although only a “caricature” of the three-dimensional helium atom, is realistic enough to capture some of the most important physical features of the helium atom.