In this chapter we turn to the formulation of the theory of the interaction of intense laser fields with atoms in the important case where the laser photon energy is much smaller than the ionization potential of the initial atomic state. When the intensity is sufficiently high and the frequency sufficiently low, ionization proceeds as if the laser electric field were quasi-static. In this regime, it is appropriate to make the “strong-field approximation,” or SFA, in which one assumes that an active electron, after having been ionized, interacts only with the laser field and not with its parent core. Using this approximation, Keldysh [1] showed that analytical expressions for the rate of ionization can be obtained when the electric-field amplitude, the laser frequency and the binding energy of the initial state are such that the Keldysh parameter γK defined by Equation (1.8) is much less than unity and the photoelectron does not escape by over-the-barrier ionization (OBI). However, the applicability of the SFA extends beyond this regime and, more importantly, it can be used to investigate high-order ATI and high-order harmonic generation. The SFA also provides a framework in which the physical origin of these processes, embodied in the semi-classical three-step recollision model introduced in Section 1.3, can be understood.

We begin in Section 6.1 by examining the low-frequency limit of the Floquet theory and showing how the total ionization rate of the atom can be obtained using the adiabatic approximation.

In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively.

We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies, which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.

The availability of intense laser fields over a wide frequency range, in the form of short pulses of coherent radiation, has opened a new domain in the study of light–matter interactions. The peak intensities of these laser pulses are so high that the corresponding laser fields can compete with the Coulomb forces in controlling the dynamics of atomic systems. Atoms interacting with such intense laser fields are therefore exposed to extreme conditions, and new phenomena occur which are known as multiphoton processes. These phenomena generate in turn new behaviors of bulk matter in strong laser fields, with wide-ranging applications.

The purpose of this book is to give a self-contained and unified presentation of high- intensity laser–atom physics. It is primarily aimed at physicists studying the interaction of laser light with matter at the microscopic level, although it is hoped that any scientist interested in laser–matter interactions will find it useful.

The book is divided into three parts. The first one contains two chapters, in which the basic concepts are presented. In Chapter 1, we give a general overview of the new phenomena discovered by studying atomic multiphoton processes in intense laser fields. In Chapter 2, the theory of laser–atom interactions is expounded, using a semi-classical approach in which the laser field is treated classically, while the atom is described quantum mechanically.

In this chapter, we shall discuss the theory of laser–atom interactions, using a semi-classical method in which the laser field is treated classically, while the atom is studied by using quantum mechanics. This semi-classical approach constitutes an excellent approximation for intense laser fields, since in that case the number of photons per laser mode is very large [1, 2]. In addition, spontaneous emission can be neglected. We begin therefore by giving in Section 2.1 a classical description of the laser field in terms of electric- and magnetic-field vectors satisfying Maxwell's equations. We start by considering plane wave solutions of these equations. Then general solutions describing laser pulses are introduced. The dynamics of a classical electron in the laser field, and in particular the ponderomotive energy and force, are discussed in Section 2.2. Neglecting first relativistic effects, we write down in Section 2.3 the time-dependent Schrödinger equation (TDSE), which is the starting point of the theoretical study of atoms in intense laser fields, and introduce the dipole approximation. In the subsequent two sections, we study the behavior of the TDSE under gauge transformations and the Kramers frame transformation. In view of the central role that the time evolution operator plays in the development of the theory of laser–atom interactions, some general properties of this operator are reviewed in Section 2.6.

One of the major non-perturbative methods used to study atoms in intense laser fields is the direct numerical integration of the wave equations describing atoms interacting with laser fields [1]. This is an attractive alternative to the methods discussed in the two preceding chapters, since solutions of the wave equations can be obtained by numerical integration for a wide range of laser intensities and frequencies. In addition, no restrictions need to be imposed on the type of laser pulses which are used, making the numerical integration of wave equations particularly useful for the study of interaction of atoms with short laser pulses.

However, the numerical integration of the wave equations is computationally very intensive, for the following reasons. Firstly, at high laser intensities, and especially for low frequencies, the ionized electrons can acquire quite high velocities and their quiver motion becomes much larger than the size of the initial atomic orbit. The corresponding wave packets can therefore travel large distances in short time intervals. As a result, the spatial grids used to follow the motion of these wave packets must be large and have small spatial separations. Secondly, the discretization of time, used in all numerical integration schemes, requires a large number of small steps in order to obtain accurate results. Thirdly, the direct numerical integration of the wave equations becomes extremely demanding for atoms with more than one active electron.

In this chapter, we focus our attention on the multiphoton ionization (MPI) of atoms interacting with intense laser fields with wavelengths in the infra-red to visible part of the spectrum. Section 8.1 is devoted to multiphoton single ionization. We begin by giving an overview of key early experiments, in particular those exploring the phenomenon of “above-threshold ionization” (ATI). We then discuss general features of ATI spectra and consider how these features can be understood within the framework of the semi-classical model. We conclude the section by examining how two-color processes can be used to study MPI. In Section 8.2, we analyze multiphoton double ionization, a process which has attracted considerable attention due to the prominent role played by electron correlation effects. Detailed reviews of atomic multiphoton ionization and ATI have been given by Joachain [1], DiMauro and Agostini [2], Protopapas, Keitel and Knight [3], Joachain, Dörr and Kylstra [4], Kylstra, Joachain and Dörr [5], Dörner et al. [6], Becker et al. [7] and Lewenstein and L'Huillier [8].

Multiphoton single ionization

As noted in Section 1.3, multiphoton ionization (MPI) was first observed in 1963 by Damon and Tomlinson [9] and also investigated in 1965 by Voronov and Delone [10] and Hall, Robinson and Branscomb [11]. In the following two decades, a number of experiments were performed to study various aspects of MPI, and results were obtained concerning the dependence of the ionization yields on the laser intensity, absolute MPI cross sections and the resonantly enhanced multiphoton ionization (REMPI) phenomenon.

This chapter is devoted to the study of harmonic generation in atoms and the physics of attosecond pulses, also called attophysics, which are two major topics in the study of high-intensity laser–atom interactions. We start in Section 9.1 by reviewing important experiments, with particular emphasis on high-order harmonic generation, which is a very interesting probe of the behavior of atoms interacting with intense laser fields. In Section 9.2, we discuss harmonic generation calculations, first at the microscopic (single-atom response) level and then at the macroscopic level. The main properties of harmonics and some of their applications are discussed in Section 9.3. Finally, in Section 9.4, we examine how attosecond pulses can be produced and used to investigate the dynamics of atoms at unprecedented time and space scales. Reviews of harmonic generation have been given by L'Huillier, Schafer and Kulander [1], L'Huillier et al. [2], Joachain [3], Salières et al. [4], Protopapas, Keitel and Knight [5], Joachain, Dörr and Kylstra [6], Brabec and Krausz [7], Salières [8, 9] and Salières and Christov [10]. Attosecond physics has been reviewed by Agostini and DiMauro [11], Scrinzi et al. [12], Kienberger et al. [13], Niikura and Corkum [14], Krausz and Ivanov [15], Lewenstein and L'Huillier [16] and Scrinzi and Muller [17].

Experiments

In this section, we give an overview of key experiments which have been performed in the field of harmonic generation and the production of attosecond pulses.

In this chapter, we shall analyze the interaction of atoms with intense laser fields whose frequency is much larger than the threshold frequency for one-photon ionization. We begin in Section 7.1 by discussing the high-frequency Floquet theory (HFFT) within the framework of the non-relativistic theory of laser–atom interactions in the dipole approximation. In Section 7.2, the HFFT is applied to study the structure of atomic hydrogen in intense, high-frequency laser fields. An interesting prediction of the HFFT is atomic stabilization, whereby the ionization rate of an atom interacting with an intense, high-frequency laser field decreases as the laser intensity increases. This phenomenon is analyzed in Section 7.3, where we discuss ionization rates obtained within the HFFT as well as from ab initio Floquet calculations. We then consider investigations of stabilization based on the direct numerical integration of the time-dependent Schrödinger equation (TDSE). Finally, we examine the influence of non-dipole and relativistic effects on atomic stabilization. Detailed reviews of the HFFT and stabilization have been given by Gavrila [1–3].

High-frequency Floquet theory

The HFFT is based on analyzing the atom–laser field interaction in the accelerated, or Kramers–Henneberger (K–H), frame [4, 5]. It was developed by Gavrila and Kaminski [6] to study electron scattering by a potential in the presence of a high-frequency laser field and generalized by Gavrila [7] to investigate the atomic structure and ionization of decaying dressed states.