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.

The discussions in the previous chapters show that already the spontaneous emission, i.e., the simplest manifestation of the quantum-optical fluctuations, exhibits highly nontrivial features as soon as the quantum light is coupled to interacting many-body systems. We have seen, for example, that pronounced resonances in the semiconductor luminescence originate from a nontrivial mixture of exciton and plasma contributions in contrast to the simple transitions between the eigenstates of isolated atomic systems. As a consequence of the conservation laws inherent to the light–matter coupling, the photon emission induces rearrangements in the entire many-body system leading, e.g., to pronounced hole burning in the exciton distribution. As discussed in Chapter 29, this depletion of the optically active excitons leads to a reduction of the total radiative recombination and the appearance of nonthermal luminescence even when the electron–hole system is in quasiequilibrium. Already these observations show that the coupled quantum-optical and many-body interactions induce new intriguing phenomena that are not explainable by the concepts of traditional quantum optics or classical semiconductor physics alone.

The foundations of quantum optics are based on systematic investigations of simple systems interacting with few quantized light modes. In this context, one can evaluate and even measure the exact eigenstates or the density matrix with respect to both the photonic and the atomic degrees of freedom. In semiconductor systems, currently the investigations using one or a few quantum dots (QD) are closest to the atomic studies because, due to their discrete eigenstates, one can treat strongly confined quantum dots to some extent like artificial atoms.

In the early 1900s more and more experimental evidence was accumulated indicating that microscopic particles show wave-like properties in certain situations. These particle-wave features are very evident, e.g., in measurements where electrons are diffracted from a double slit to propagate toward a screen where they are detected. Based on the classical averaging of particles discussed in Section 1.2.2, one expects that the double slit only modulates the overall intensity, not the spatial distribution. Experimentally, however, one observes a nearly perfect interference pattern at the screen implying that the electrons exhibit wave averaging features such as discussed in Section 2.1.2. This behavior, originally unexpected for particle beams, persists even if the experiment is repeated such that only one electron at a time passes the double slit before it propagates to the detection screen. Thus, the wave aspect must be an inherent property of individual electrons and not an ensemble effect.

Another, independent argument for the failure of classical physics is that the electromagnetic analysis of atoms leads to the conclusion that the negatively charged electron(s) should collapse into the positively charged ion because the electron–ion system loses its energy due to the emission of radiation. As we will see, this problem can be solved by including the wave aspects of particles into the analysis. In particular, as discussed in Section 2.3.3, waves can never be localized to a point without increasing their momentum and energy beyond bounds.

In this book, we encounter the hierarchy problem when we apply the equation-of-motion technique to analyze the quantum dynamics of the coupled light–matter system. To obtain systematic approximations, we use the so-called cluster-expansion method where many-body quantities are systematically grouped into cluster classes based on how important they are to the overall quantum dynamics. With increasing complexity, the clusters contain

1. independent single particles (singlets),

2. interacting pairs (doublets),

3. three-particle terms (triplets),

4. coupled four-particle contributions (quadruplets),

5. as well as higher-order correlations.

In this context, the N-particle concept is somewhat formal because it refers to generic N-particle expectation values 〈N〉, which may consist of an arbitrary mixture of carrier, photon, and phonon operators, as discussed in Section 14.2. In order to truncate the hierarchy problem at a given level, 〈N〉 is approximated through a functional structure that includes all clusters up to the predetermined level while all remaining clusters with a higher rank are omitted. It is natural that the corresponding approximations can be systematically improved by increasing the number of clusters included.

To the best of our knowledge, the idea of coupled-clusters approaches was first formulated by Fritz Coester and Hermann Kümmel in the 1950s to describe nuclear many-body phenomena. The approach was then modified for the needs of quantum chemistry by Jiri Cizek 1966 to deal with many-body phenomena in atoms and molecules. Currently, it is one of the most accurate methods to compute molecular eigenstates.

Historically, the scientific exploration of new phenomena has often been guided by systematic studies of observations, i.e., experimentally verifiable facts, which can be used as the basis to construct the underlying physical laws. As the apex of the investigations, one tries to identify the minimal set of fundamental assumptions – referred to as the axioms – needed to describe correctly the experimental observations. Even though the axioms form the basis to predict the system's behavior completely, they themselves have no rigorous derivation or interpretation. Thus, the axioms must be viewed as the elementary postulates that allow us to formulate a systematic description of the studied system based on well-defined logical reasoning. Even though it might seem unsatisfactory that axioms cannot be “derived,” one has to acknowledge the paramount power of well-postulated axioms to predict even the most exotic effects. As is well known, the theory of classical mechanics can be constructed using only the three Newtonian axioms. On this basis, an infinite variety of phenomena can be explained, ranging from the cyclic planetary motion all the way to the classical chaos.

In this book, we are mainly interested in understanding how the axioms of classical and quantum mechanics can be applied to obtain a systematic description for the phenomena of interest. Especially, we want to understand how many-particle systems can be modeled, how quantum features of light emerge, and how these two aspects can be combined and utilized to explore new intriguing phenomena in semiconductor quantum optics.