Introduction

The present chapter is about models. All the models are inspired by some physical system: sandpiles, earthquakes, magnetic vortex motion, forest fires, interface growth, or biological evolution. The models are defined in terms of a dynamical variable – for example, the local slope of the sand heap or the stress in the earthquake fault. The dynamical variable or field is updated in every time step according to some algorithm. The choice of the updating algorithm is, to some degree, arbitrary. The criteria for choosing the relevant definitions are, for the most part, simplicity and intuition. Statistical mechanicians have this overall belief that complexity arises from simplicity: that the intricate and complex behavior found in many systems is due to the large number of degrees of freedom, rather than caused by some very complicated behavior of the individual degrees of freedom. All the models described in this chapter are formulated according to this paradigm. In addition, they are all designed with an eye toward numerical ease and efficiency when simulated on computers.

For ease of presentation, for each model we shall first simply present the definition of the model and the conclusions derived from numerical studies thereof. Since the aim is to understand the world around us, we shall use separate sections to discuss the relevance and shortcomings of each model. As always when building models of Nature, one proceeds through steps of increasing refinement, and so also for models of SOC systems.

We discuss here some details of writing a computer code for the dynamics of the lattice gas defined in Section 4.4. The code is not optimized; it is included as a guide and for inspiration only.

The configuration of the particles on the (two-dimensional) lattice is stored in the array n(l: L, 1: L). The elements of this array take the value n(i, j) =1 if the site (i, j) is occupied by a particle; otherwise, n(i, j) =0. The update, as defined in Section 4.4.1, can be coded as sketched below. A complete code of the lattice gas will include an initiation of the array n(i, j), and special attention must be paid to the boundary. Since the lattice is updated simultaneously, we need the array nnew (1: L, 1: L) to store intermediate changes in the particle configuration. We update the lattice by visiting all the empty sites of the lattice. An empty site (i, j) becomes an occupied site if a particle on one of the eight neighbor sites is able to move onto site (i, j). We therefore visit all eight neighbor sites to calculate the forces acting on particles in these sites. We then check if one of these particles has a force pointing toward site (i, j) and is of a magnitude greater than any other force pointing toward site (i, j).

The nature of the critical state is described by the response of a system to external perturbation. For systems exhibiting noncritical behavior, the reaction of the system is described by a characteristic response time and characteristic length scale over which the perturbation is felt spatially. Although the response of a noncritical system may differ in detail as the system is perturbed at different positions and at different times, the distribution of responses is narrow and is well described by the average response. For a critical system, the same perturbation applied at different positions or at the same position at different times can lead to a response of any size. The average may not be a useful measure of the response; in fact, the average might not even exist.

Response Distributions

To illustrate the description of the critical state, consider the sandpile. We probe the state by adding one single grain of sand to a (randomly) chosen position on the slope. The extra grain will induce an avalanche characterized by such spatial and temporal measures as the total number s of sand grains involved in the avalanche and the lifetime t of the avalanche. We denote the statistical distributions describing the response by P(s) and P(t).

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.

In Chapter 1 we discussed some concepts of chaos, its manifestations and applications on an introductory level from a purely qualitative point of view. The concepts were introduced ad hoc and in a pictorial manner. We will now turn to a more detailed investigation of chaos in order to prepare the tools and concepts needed for the discussion of chaotic atomic and molecular systems.

For a long time researchers thought that every given nonlinear system required its own individual method of solution. If this were the case, there could not be any general theory of nonlinear systems. Rather, the science of nonlinear systems would resemble descriptive sciences such as 19th century biology or geology. The best one could offer would be a catalogue of nonlinear systems together with their individual properties and methods of solution (if any). Luckily, the situation is much more promising. Not so long ago it was shown by Feigenbaum (1978, 1979) that there is universality in chaos. Universality is the key property of chaos. Universality means that all nonlinear (chaotic) systems can be analysed using a common set of methods and tools. Thus, a given nonlinear system does not require special treatment. It is always amenable to a general analysis, whose elements are discussed in the following sections.

It was Poincaré who introduced a major revolution in the analysis of dynamical systems in classical mechanics.