At the conceptual level, the quantization procedure replaces the pure particle description of classical mechanics by a particle-wave treatment satisfying the requirements of the fundamental wave–particle dualism. Technically, one can always start from a generalized wave equation and use it as a basis to discuss wave-front propagation in the form of rays following classical particle trajectories, see Sections 2.2.1 and 3.1. Consequently, any wave theory inherently contains that aspect of the wave–particle dualism.

Especially, Maxwell's equations of electrodynamic theory include both wave and particle aspects of electromagnetic fields. Hence, it could be that there are no additional “quantum,” i.e., wave–particle dualistic aspects. However, as we discuss in this chapter, Maxwell's equations can be expressed in the framework of Hamilton's formulation of particle mechanics. This then suggests that in addition to the approximate ray-like behavior, electromagnetic fields do have a supplementary level of particle-like properties.

Due to the fundamental axiom of wave–particle dualism, the additional particle-like aspects of electromagnetic fields must then also have a wave counterpart that can be axiomatically introduced via the canonical quantization scheme described in Section 3.2.3. Since the spatial dependency in Maxwell's equation contains wave–particle dualism already at the classical level, it is clear that the new level of quantization cannot be a simple real-space feature but has to involve some other space describing the structure of the electromagnetic fields. In fact, this quantization yields several very concrete implications such as quantization of the light energy, elementary fluctuations in the light-field amplitude and intensity, and more.

The optically generated excitations in semiconductors constitute a genuine many-body system. To describe its quantum-optical features, we have to expand significantly the theoretical models used so far. However, the important insights of Chapters 16–23 are already presented in a form in which most of them can directly be used and generalized to analyze central properties of the optical excitations in solids. As for atoms, the optical transitions in semiconductors are induced via dipole interaction between photons and electrons. We can thus efficiently construct a systematic quantum-optical theory for semiconductors by following the cluster-expansion approach.

One of the main differences from atoms is that the electronic excitations in semiconductors form a strongly interacting many-body system. Thus, we must systematically treat the arising Coulomb-induced hierarchy problem together with the quantum-optical one. Moreover, the coupling of electrons to lattice vibrations, i.e., the phonons, produces yet another hierarchy problem. In addition, in solid-state spectroscopy one often uses multimode light fields such that one cannot rely on the single-mode simplifications to study semiconductor quantum optics.

As shown in Chapter 15, the Coulomb-, phonon-, and photon-induced hierarchy problems have formally an identical structure. Thus, we start the analysis by investigating how semiconductor quantum optics emerges from the dynamics of correlated clusters. We first focus on the basic properties of the optical transitions in the classical regime. This means investigating the fundamental optical phenomena resulting from the singlets. The full singlet–doublet approach is presented in Chapters 28–30.

A wide variety of quantum-optical effects can be understood by analyzing atomic model systems interacting with the quantized light field. Often, one can fully calculate and even measure the quantum-mechanical wave function and its dependence on both the atomic and the light degrees of freedom. By elaborating on and extending this approach, researchers perpetually generate intriguing results and new insights allowing for the exploration and utilization of effects encountered only in the realm of quantum phenomena.

By now, quantum-optical investigations have evolved from atoms all the way to complex systems, such as solids, in particular semiconductors. As a profound conceptual challenge, the optical transitions in semiconductors typically involve an extremely large number of electronic states. Due to their electric charge, the optically active electrons experience strong Coulomb interaction effects. Furthermore, they are coupled to the lattice vibrations of the solid crystal. For such an interacting many-body system, the overwhelmingly large number of degrees of freedom makes it inconceivable to measure the full wave function; we obviously need new strategies to approach semiconductor quantum optics. The combination of quantum-optical and many-body interactions not only leads to prominent modifications of the effects known from atomic systems but also causes new phenomena without atomic counterparts.

In this book, we develop a detailed microscopic theory for the analysis of semiconductor quantum optics. As central themes, we discuss how the quantum-optical approach can be systematically formulated for solids, which new aspects and prospects arise, and which conceptual modifications have to be implemented.

Chapters 7–9 present the classical description of many-body systems in a way that allows us to identify the canonical variables for the coupled system of matter and electromagnetic fields. Thus, we are now in the position to apply the canonical quantization scheme outlined in Section 3.2.3. We already know that the quantization extends the particle concept to include also wave aspects such that the overall description satisfies the wave–particle duality. Once both matter and light are quantized, we have a full theory which can be applied to treat many interesting phenomena in the field of semiconductor quantum optics.

The quantization is conceptually more challenging for light than for particles because Maxwell's equations already describe classical waves. However, the mode expansion for the vector potential and the generalized transversal electric field allows us to identify the particle aspects associated with light waves. This approach presents the system dynamics in the form of classical Hamilton equations for the mode-expansion coefficients. Thus, the canonical quantization deals with these coefficients and supplements an additional wave character to them. In other words, the light quantization introduces complementarity at several levels: classical light is already fundamentally a wave while its dualistic particle aspects emerge in ray-like propagation, as discussed in Chapter 2. At the same time, the mode expansion identifies additional particle aspects and the quantization of the mode-expansion coefficients creates a new level of wave–particle dualism. In this chapter, we apply the canonical quantization scheme to derive the quantized system Hamiltonian.

In the previous chapters, we have seen that quantum-optical correlations can produce effects that have no classical explanation. In particular, the matter excitations can depend strongly on the specific form of the quantum fluctuations, i.e., the quantum statistics of the light source. For example, Fock-state sources can produce quantum Rabi flopping with discrete frequencies while coherent-state sources generate a sequence of collapses and revivals in atomic excitations. Hence, not only the intensity or the classical amplitude of the field is relevant but also the quantum statistics of the exciting light influences the matter response. Even if we take sources with identical intensities, the resulting atomic excitations are fully periodic for a Fock-state excitation while a coherentstate excitation produces a chaotic Bloch-vector trajectory with multiple collapses and revivals.

In this chapter, we use this fundamental observation as the basis to develop the concept of quantum-optical spectroscopy. We show in Chapter 30 that this method yields a particularly intriguing scheme to characterize and control the quantum dynamics in solids. Since one cannot imagine how to exactly compute the many-body wave function or the density matrix, we also study how principal quantum-optical effects can be described with the help of the cluster-expansion scheme.

Quantum-optical spectroscopy

Historically, the continued refinement of optical spectroscopy and its use to manipulate the states of matter has followed a very distinct path where one simultaneously tries to control and characterize light with increased accuracy.

As discussed in the previous chapter, we adopt the Coulomb gauge for all our further investigations starting from the many-body Hamiltonian (8.86) and the mode expansion (8.87). Before we proceed to quantize the Hamiltonian, we want to make sure that our analysis is focused on the nontrivial quantum phenomena. Thus, we first have to identify and efficiently deal with the trivial parts of the problem.

Often, the experimental conditions are chosen such that only a subset of all the electrons in a solid interacts strongly with the transversal electromagnetic fields while the remaining electrons and the ions are mostly passive. To describe theoretically such a situation in an efficient way, it is desirable to separate the dynamics of reactive electrons from the almost inert particles that merely produce a background contribution. This background can often be modeled as an optically passive response that is frequency independent and does not lead to light absorption.

In this chapter, we show how the passive background contributions can be systematically identified and included in the description. As the first step, we introduce the generalized Coulomb gauge to eliminate the scalar potential and to express the mode functions and the canonical variables. This leads us to a new Hamiltonian with altered Coulomb potential and mode functions. This generalized Hamiltonian allows us to efficiently describe optically active many-body systems in the presence of an optically passive background.