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.

The major portion of this monograph discusses how given pulses of laser radiation affect individual atoms. This chapter inverts that relationship, describing how the matter alters the fields. Those field changes provide quantitative measures of the quantum-state changes produced in the atoms. Their description therefore is an adjunct to Chap. 19.

Basically the incident radiation produces excitation which, in turn, alters the various multipole moments of the atoms. When viewed as a macroscopic sample of matter, such changes alter the electric polarization field P and the magnetization field M of the matter through which the radiation must pass. The Maxwell equations, see App. C.1, provide the needed description of how the P and M fields alter the electric and magnetic fields E and B. The combination of the Maxwell equations for the fields and the Schrödinger, Bloch, or Liouville equations for the atoms provide the tools needed to construct a self-consistent description of radiation passing through matter – atoms responding to a pulsed field and traveling waves being modified by the resulting atomic changes [All87; Ale92; Muk99; Die06; Vit01a]. The present chapter, drawing on [Sho90, Chap. 12] and App. C, discusses this theory.

Incoherent radiation passing through matter typically undergoes exponential attenuation in accord with eqn. (6.9). A measure of the incremental change of intensity in distance L by absorption coefficient κ is the optical depth κL. This parameter appears in rateequation treatments of incoherent light attenuation.

The nature of measurements, and their place within quantum theory, has engaged physicists and philosophers for generations [Bra92; Sch03]. Much of that interest centered on variables such as position and momentum of free particles, whose values form a continuum. The present monograph deals with discrete quantum states; the measurements are those required to specify as completely as possible a particular discrete quantum state Ψ or, more generally, a density matrix ρ defined within a finite-dimensional Hilbert space.

General remarks

General system. At the outset we assume that the possible quantum states are a small number – the N essential states used in formulating the time-dependent Schrödinger equation or specifying the dimensions of the density matrix. To completely characterize a density matrix for such a system we require the N2 elements. Of these the N diagonal elements are real valued, while the off-diagonal elements of the upper right side are complex conjugates of those on the lower left. Thus with allowance for the requirement of unit trace a total of N2 – 1 real numbers suffice to completely specify the density matrix. These values must be consistent with the constraints discussed in Sec. 16.6.3.

Pure state. If it is known that the system is in a pure state, we require the magnitude and phase of N probability amplitudes. These are constrained by normalization, and so only 2N –1 real numbers are needed. Out of these 2N – 1 parameters the overall phase of the statevector is usually not of interest (but see Chap. 20).

As light passes an atom, it exerts forces on the charges, electrons, and nuclei that alter the atomic structure. These may be slight distortions (perturbations) of the electron cloud or they may be more severe, as described in subsequent sections of this monograph. We wish to determine those changes, given the radiation field, or to devise a radiation field that will produce specified changes.

The changes to the atomic structure also affect the radiation that subsequently passes the atom. To describe those effects we must consider wave equations for radiation in the presence of altered atomic structure. More generally we must find self-consistent equations for the atoms and the field together, as discussed in Chaps. 21 and 22. Here we consider mathematical descriptions of the influence of coherent radiation on individual atoms, molecules, or other single quantum systems.

Individual atoms

Traditional sources of emission and absorption spectra, though revealing energy states of the constituent atoms and molecules, are macroscopic samples. One observes averaged characteristics of many individual particles, see Chap. 16. Quantum theory offers the basic formalism for dealing with individual atoms exposed to controlled radiation fields. Several experimental techniques provide acceptable approximations to this ideal. The following paragraphs note some of these examples.

Vapors

Neutral atoms or molecules in a vapor move freely along straight-line paths, interrupted by brief collisions that redirect the two collision partners. When the kinetic energies of the two partners are small, there can be no transfer of kinetic energy into internal energy of either particle – the collision is elastic.

Quantum changes of three-state systems have some similarities with those of two-state systems. When subject to steady illumination the populations may undergo oscillations similar to the Rabi cycling of two-state systems, and various forms of adiabatic following are possible. Analytic solutions to the relevant TDSE exist [Sho90, Chap. 23]. The additional degree of freedom, typically allowing controllable parameters of a second laser pulse, allows a wider variety of controlled excitation. The resulting differences and similarities to two-state systems have been discussed at length [Whi76; Sho77; Rad82b; Yoo85; Car87].

Three-state linkages

Two-field linkages. The simplest extension of two-state excitation allows two laser fields, here identified by letters P (for pump) and S (for Stokes), as befits the stimulated Raman process discussed in Chap. 14. The carrier frequencies of the two fields, ωP and ωS, are each assumed to be close to resonance with one, and only one, Bohr frequency, so that each field can be uniquely identified with a particular transition (failure of this restriction, and the resulting linkage ambiguity, is discussed in [Una00]). I will assume that the P field is (near) resonance only with the 1–2 transition, while the S field is (near) resonant only with the 2–3 transition; these interactions thereby form a two-step linkage chain. This system has three possible linkage patterns, shown in Fig. 13.1.

The linkage patterns (sometimes called configurations) differ by the ordering of the energies of the linked states.With the assumption that population initially occupies state 1, as in Fig. 13.1, the definitions are:

Ladder: The ladder system has the energy ordering E1 < E2 < E3.

Manipulation of the internal structure of atoms and molecules – altering the quantum states of submicroscopic systems – makes an increasingly significant contribution to contemporary technology, as electronic circuits continue to shrink in size and new opportunities appear for applying abstract quantum theory to the creation of practical devices. The structural changes range from simple perturbative distortions of the electronic charge distribution to complete transformation into an excited energy state or the creation of superposition states whose properties cannot be fully described without quantum theory.

This monograph discusses ways of inducing such changes, primarily (but not only) with pulses of laser light, and ways of picturing the changes with the aid of suitable mathematical tools. Aiming at a level suitable for advanced undergraduates or researchers it explains the basic principles that underly the quantum engineering of devices used for such applications as coherent atomic excitation and quantum information processing.

Presupposing some familiarity with quantum mechanics, it first introduces notions of atoms (or other localized quantum systems) and quantum states, and of radiation (specifically laser pulses), defining thereby the essential observable quantities with which theory must deal. It presents the constructs – probabilities, probability amplitudes, wavefunctions, and statevectors – that serve as variables for the mathematics. It then discusses the differential equations that describe laser-induced changes to atomic structure. It contrasts the pre-laser incoherent absorption of energy, governed by rate equations, with the coherent regime of laser-induced changes, governed by the time-dependent Schrödinger equation that is the foundation for all descriptions of quantum-mechanical changes.

Between every pair of quantum states for which a nonzero electric or magnetic multipole moment exists there can take place a radiative transition. When the quantum states are both discrete, as they are for pairs of bound states, the radiation is discrete – a spectral line. The frequency of that spectral line is set by the difference of the two energies, and can occur in any region of the electromagnetic spectrum. If there is available a source of coherent radiation at that frequency, then coherent quantum-state manipulation is possible.

For many years spectroscopists routinely assembled collections of wavelengths and line strengths (or transition probabilities) for various elements and molecules. The National Bureau of Standards (NBS), now the National Institute for Standards and Technology (NIST), collected, organized, and published much of this data. Their website, www.nist.gov/pml/data/handbook/index.cfm, provides ready access to this information for all the elements and many molecules. Much of this data has appeared in the Journal of Physical and Chemical Reference Data, published by the American Institute of Physics (AIP). Diagrams showing the relative positions of energies and the connecting transitions, often called Grotrian diagrams [Bas75; Moo68; Lan99], are helpful for presenting the excitation linkages.

Spectroscopic parameters

From traditional spectroscopic studies come several parameters with which to describe the resonant interaction between light and an atom – one that exists, ideally, in free space and which therefore has degenerate energy levels.