Table of contents

We review basic quantum electrodynamics and quantum optics aspects in microstructures that exhibit a gap in the spectrum of the electromagnetic radiation they support, known as photonic crystals. After a brief sketch of the properties of such materials we discuss the behaviour of few-level atoms or collections thereof with transition frequencies inside and in the vicinity of the gap. The discussion is cast in terms of a unified formalism which facilitates the comparison with standard cavity-atom physics.

Calorimetry plays a crucial role in modern experimental physics. Calorimeters are essential tools to extract physics in accelerator and non-accelerator experiments. The physics phenomena at the base of cascading processes in matter and the basic principles of calorimeters operation are reviewed at the light of data obtained from running experiments or from sets of dedicated measurements and the constraints from physics requirements. From this understanding comes the possibility of building powerful calorimetric systems with the optimal performances required by future experiments at high-energy and ultrahigh-energy regimes.

A comprehensive introduction to two-dimensional conformal field theory is given. The structure of the meromorphic subtheory is described in detail, and a number of examples are presented explicitly. Standard constructions such as the coset and the orbifold construction are explained. The concept of a representation of the meromorphic theory is introduced, and the role of Zhu's algebra in classifying highest weight representations is elucidated. The fusion product of two representations and the corresponding fusion rules are defined, and Verlinde's formula is explained. Finally, higher correlation functions are considered, and the polynomial relations of Moore and Seiberg and the quantum group structure of chiral conformal field theory are discussed. The treatment is relatively general and also allows for a description of less well known classes of theories such as logarithmic conformal field theories.

The quantum dynamics of mesoscopic or macroscopic systems is always complicated by their coupling to many `environmental' modes. At low Tthese environmental effects are dominated by localizedmodes, such as nuclear and paramagnetic spins, and defects (which also dominate the entropy and specific heat). This environment, at low energies, maps onto a `spin bath' model. This contrasts with `oscillator bath' models (originated by Feynman and Vernon) which describe delocalizedenvironmental modes such as electrons, phonons, photons, magnons, etc. The couplings to Nspin bath modes are independentof N(rather than the ~O(1/(N )<sup>1/2</sup> ) dependence typical of oscillator baths), and often strong. One cannot in general map a spin bath to an oscillator bath (or vice versa); they constitute distinct `universality classes' of quantum environment.

We show how the mapping to spin bath models is made, and then discuss several examples in detail, including moving particles, magnetic solitons, nanomagnets, and SQUIDs, coupled to nuclear and paramagnetic spin environments.

We then focus on the `central spin' model, which couples a central two-level system to a background spin bath. It is the spin bath analogue of the famous `spin-boson' oscillator model, and describes, e.g., the tunnelling dynamics of nanoscopic and mesoscopic magnets and superconductors. We show how to average over (or `integrate out') spin bath modes, using an operator instanton technique, to find the central spin dynamics. The formal manouevres involve four separate averages - each average corresponds physically to a different `decoherence' mechanism acting on the central spin dynamics. Each environmental spin has its own topological `spin phase', which by interacting with the phase of the central system, decoheres it - this can happen even without dissipation. We give analytic results for the central spin correlation functions, under various conditions.

We then describe the application of this theory to magnetic and superconducting systems. Particular attention is given to recent work on tunnelling magnetic macromolecules, where the role of the nuclear spin bath in controlling the tunnelling is very clear; we also discuss other magnetic systems in the quantum regime, and the influence of nuclear and paramagnetic spins on flux dynamics in SQUIDs.

Finally, we discuss decoherence mechanisms and coherence experiments in superconductors and magnets. We show that a spin bath environment causes decoherence even in the T 0 limit. Control of this decoherence will be essential in the effort to construct `qubits' for quantum computers.

Unfortunately figures 37 and 42 in our review have been interchanged. The figure appearing above the caption for figure 37 on page 1625 is figure 42, and should appear above the caption for figure 42 on page 1630, and vice versa.