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.

The experimental investigation of chaos in atomic physics began with the historic experiment on microwave ionization of Rydberg atoms reported by Bayfield and Koch in 1974. The central result is the existence of an ionization threshold as a function of the microwave field. At the time (1974) this result was totally unexpected since ionization thresholds, in analogy to the photo-electric effect, were thought to appear only as a function of the frequency. Nowadays, especially in the light of the material presented in Chapters 5 and 6, the existence of an ionization threshold in the microwave field is less surprising and may be attributed to the existence of a critical microwave field that marks the onset of global chaos in the classical analogue of the Bayfield-Koch experiments. But at the time the Bayfield-Koch experiment was conducted, a connection with chaos was not suspected. Leopold and Percival (1978, 1979) were the first to investigate the Bayfield-Koch experiments using purely classical mechanics. This is allowed on the basis of the correspondence principle, Leopold and Percival argued, since the quantum numbers involved in the ionization experiments are large. This line of thought turned out to be very fruitful. With the help of Monte Carlo simulations of the time evolution of classical trajectories in phase space, Leopold and Percival were able to reproduce the existence and location of the microwave thresholds established by the Bayfield-Koch experiment.