Classical and Quantum Optics of Semiconductor Nanostructures

Abstract

Optical properties of semiconductor nanostructures are widely studied both experimentally and theoretically. They are interesting from an application point of view while they also provide an ideal playground to study Coulomb effects, light–matter interaction, and so forth. For the theoretical modeling, the strongly interacting charge carriers inside a semiconductor present a considerable challenge. This is intensified if also the electromagnetic radiation and potentially also the lattice vibrations have to be treated quantum mechanically. Direct solutions of, e.g., the Schrodinger equation are completely out of question, and a successful theoretical approach has to find consistent methods of truncating the infinite hierarchy problem caused by the interaction. In particular, Coulomb correlations have to be dealt with on the same footing as phonon or photon correlations. Our theoretical approach is based on the Heisenberg equation of motion where the precise density matrix of the total system never has to be known. Instead, we will show in this article how quantum mechanically correct equations of motion can be derived for any quantities of interest as soon as the total system Hamiltonian is known. Thus, the precise knowledge of the Hamilton operator is of utmost importance and it should therefore include all relevant interaction mechanisms of all interacting quasi-particles of interest. Due to this prominent role of the Hamiltonian, we have split this article into two parts. The first two sections deal exclusively with the derivation of the semiconductor Hamiltonian of a nanostructure interacting with both a quantized light field and quantized lattice vibrations. While Section 10.2 deals with the contributions of the non-interacting quasi-particles and introduces important concept of the electronic band structure, the interaction contributions are discussed in Section 10.3. In Section 10.4, we calculate the elementary Heisenberg equation