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.

What happens if one sprinkles electrons onto the surface of liquid helium? Surprisingly the electrons are not absorbed into the bulk of the fluid, but form a quasi-two-dimensional sheet of electrons concentrated at some distance above the helium surface. In general, electrons hovering above the surface of a dielectric are called surface state electrons. An excellent review of surface state electrons is that by Cole (1974).

Surface state electrons are especially interesting in the context of chaos and quantum chaos. This is so because driven by strong microwave fields, their classical dynamics shows a transition to chaos. The investigation of microwave-driven surface state electrons as a testing ground for quantum chaos was first proposed by Jensen in 1982. So far, and to the best of our knowledge, a successful surface-state-electron (SSE) microwave ionization experiment was never carried out in the chaotic regime. This is mainly due to the formidable experimental difficulties in controlling the fragile SSE system. Electric stray fields, residual helium vapour pressure and interactions with the quantized surface modes (“ripplons”) of the liquid helium substrate make it very difficult to reach the high quantum numbers necessary for a quantum chaos experiment. It was, however, realized early on (see, e.g., Shepelyansky (1985)) that the dynamics of low angular momentum hydrogen Rydberg atoms is very similar to the dynamics of surface state electrons. Therefore, building on the knowledge accumulated in the field of surface state electrons, the focus of research shifted to the investigation of microwave-driven hydrogen and alkali Rydberg atoms.

Einstein (1917) appreciated early on that within the “old” pre-1925/26 quantum mechanics absence of integrability is a serious obstacle for the quantization of classical systems. Therefore, in retrospect not surprisingly, the quantization problem was not adequately solved until the advent of the “new” quantum mechanics by Heisenberg, Born, Jordan and Schrödinger. The new quantum mechanics did not rely at all on the notion of classical paths, and this way, unwittingly, sidestepped the chaos problem. Within the framework of the new theory, any classical system can be quantized, including classically chaotic systems. But while the quantization of integrable systems is straightforward, the quantization of classically chaotic systems, even today, presents a formidable technical challenge. This is especially true for quantization in the semiclassical regime, where the quantum numbers involved are large. In fact, efficient semiclassical quantization rules for chaotic systems were not known until Gutzwiller (1971, 1990) intoduced periodic orbit expansions. Gutzwiller's method is discussed in Section 4.1.3 below. It is important to emphasize here that the existence of chaos in certain classical systems in no way introduces conceptual problems into the framework of modern quantum theory, although, let it be emphasized again, chaos came back with a vengeance from the “old” days of quantum mechanics. Even given all the modern day computer power accessible to the “practitioner” of quantum mechanics, chaos is the ultimate reason for the slow progress in the numerical computation of even moderately excited states in such important, but chaotic problems as the helium atom.

In this and the following chapters we will encounter various time dependent and time independent atomic physics systems whose classical counterparts are chaotic. All the systems discussed in the remaining chapters of this book are examples of type I systems, i.e. examples for quantized chaos. This is natural since quantized chaos is by far the most important type of quantum chaos relevant in atomic and molecular physics. In the category “time dependent systems”, we discuss the rotational dynamics of diatomic molecules (Section 5.4), the microwave excitation of surface state electrons (Chapter 6), and hydrogen Rydberg atoms in strong microwave fields (Chapters 7 and 8). All these systems are driven by externally applied microwave fields. For strong fields none of these three systems can be understood on the basis of quantum perturbation theory, as the involved multi-photon orders are very high. Processes of multiphoton orders 100 to 300, typically, have to be considered. It is important to realize that in this day and age, with powerful super-computers at hand, there is no problem in implementing a perturbation expansion of such high orders. But the emphasis is on understanding the processes involved. Although multi-photon perturbation theory provides valuable insight into the physics of low order multi-photon processes important in the case of weak applied fields (an example is discussed in Section 6.3), not much insight is gained from a perturbation expansion that has to be carried along to the 100th order and beyond in order to converge.

In all of the atomic and molecular systems studied in the previous chapters the relevant dynamics was the bound-space dynamics with the continuum playing either no role at all (see, e.g., the kicked rotor and the driven CsI molecule), or only an auxiliary role for probing the bound-space dynamics with the help of the observed ionization signal (see, e.g., the driven surface state electrons and microwave-driven hydrogen atoms). In this chapter we focus on atomic and molecular scattering, i.e. on processes in which the continuum plays an essential role. This subject has recently attracted much attention as dynamical instabilities and chaos have been discovered in the simplest scattering systems. Complicated scattering in an atomic physics system was noticed as early as 1971 by Rankin and Miller in the theoretical description of a simple chemical reaction. In 1983 Gutzwiller observed complicated behaviour of the quantum phase shift in a schematic model of chaotic scattering. 1986 saw the publication of various important papers on chaotic scattering. Eckhardt and Jung (1986) reported on the occurrence of chaos in a model scattering system. Chaos was found by Davis and Gray (1986) in the classical dynamics of unimolecular reactions, and Noid et al. (1986) noticed fractal behaviour in the He – I2 scattering system. These papers were an important catalyst for the creation of a whole new field: chaotic scattering.

Poincaré (1892, reprinted (1993)) was the first to appreciate that exponential sensitivity in mechanical systems can lead to exceedingly complicated dynamical behaviour. Surprisingly, complicated systems are not necessary for chaos to emerge. In fact, chaos can be found in the simplest dynamical systems. Well known examples are the driven pendulum (Chirikov (1979), Baker and Gollub (1990)), the double pendulum (Shinbrot et al. (1992)), and the classical versions of the hydrogen atom in a strong magnetic (Friedrich and Wintgen (1989)) or microwave (Casati et al. (1987)) field.

In general it is not possible to understand the spectra and wave functions of highly excited atoms and molecules without reference to their classical dynamics. The correspondence principle, e.g., assumes knowledge of the classical Hamiltonian as a starting point. Since the Lagrangian and Hamiltonian formulations of classical mechanics provide the most natural bridge to quantum mechanics, we start this chapter with a brief review of elementary concepts in Lagrangian and Hamiltonian mechanics (see Section 3.1). The double pendulum, an example of a classically chaotic system, is investigated in Section 3.2. This is also the natural context in which to introduce the idea of Poincaré sections. With the help of Poincaré sections we can reduce the continuous motion of a mechanical system to a discrete mapping. This is essential for visualization and analysis of a chaotic system. A discussion of integrability and chaos in Section 3.3 concludes Chapter 3.

In Chapters 5 – 7 we studied the onset of global chaos and its various manifestations in atomic and molecular systems. It was shown that in the kicked molecule (Section 5.4) the onset of chaos is responsible for population transfer to highly excited rotational states. A similar effect is active in microwave-driven surface state electrons and hydrogen Rydberg atoms where the onset of chaos results in strong ionization. But so far the focus has been on the computation of critical strengths and control parameters, whereas the ionization signal was reduced to play a secondary role as a probe, or an indicator for the onset of chaos. In this chapter we shift the focus to the investigation of the ionization signal itself, especially its time dependence.

The time dependence of weakly ionizing systems that are well described by a multi-photon process of order p has been studied extensively in the literature. In this case the time dependence of the ionization signal does not offer any surprises. We expect exponential decay with a decay rate ρ that is proportional to the pth power of the field intensity I according to ρ ∼ Ip. This prediction of multi-photon theory has been verified in numerous experiments. In fact, experimentalists often use the field dependence of the ionization rates to assign a multi-photon order to an experimentally observed ionization signal.

The purpose of this chapter is to discuss briefly, and as far as we are aware of it, the present status of research on chaos in atomic physics including trends and promising research directions. Given the enormous and rapidly growing volume of literature published every year, we cannot provide within the scope of this chapter a complete overview of existing published results. The best we can do is to select – in our opinion – representative results that indicate the status and trends in the field of chaos in atomic physics.

In Section 11.1 we discuss recent advances in quantum chaology, i.e. the semiclassical basis for the analysis of atomic and molecular spectra in the classically chaotic regime. In Section 11.2 we discuss some recent results in type II quantum chaos within the framework of the dynamic Born-Oppenheimer approximation. Recent experimental and theoretical results of the hydrogen atom in strong microwave and magnetic fields are presented in Sections 11.3 and 11.4, respectively. We conclude this chapter with a brief review of the current status of research on chaos in the helium atom.

Quantum chaology

Quantized chaos, or quantum chaology (see Section 4.1), is about understanding the quantum spectra and wave functions of classically chaotic systems. The semiclassical method is one of the sharpest tools of quantum chaology. As discussed in Section 4.1.3 the central problem of computing the semiclassical spectrum of a classically chaotic system was solved by Gutzwiller more than 20 years ago.

By now the “chaos revolution” has reached nearly every branch of the natural sciences. In fact, chaos is everywhere. To name but a few examples, we talk about chaotic weather patterns, chaotic chemical reactions and the chaotic evolution of insect populations. Atomic and molecular physics are no exceptions. At first glance this is surprising since atoms and molecules are well described by the linear laws of quantum mechanics, while an essential ingredient of chaos is nonlinearity in the dynamic equations. Thus, chaos and atomic physics seem to have little to do with each other. But recently, atomic and molecular physicists have pushed the limits of their experiments to such high quantum numbers that it starts to make sense, in the spirit of Bohr's correspondence principle, to compare the results of atomic physics experiments with the predictions of classical mechanics, which, for the most part, show complexity and chaos. The most striking observation in recent years has been that quantum systems seem to “know” whether their classical counterparts display regular or chaotic motion. This fact can be understood intuitively on the basis of Feynman's version of quantum mechanics. In 1948 Feynman showed that quantum mechanics can be formulated on the basis of classical mechanics with the help of path integrals. Therefore it is expected that the quantum mechanics of an atom or molecule is profoundly influenced, but of course not completely determined, by the qualitative behaviour of its underlying classical mechanics.