Regarding the positive aims of this course, they are: to discuss how to model the quantum behaviour of coupled light and matter; to introduce some simple models that can be used to describe such systems; to dis- cuss methods for open quantum systems that arise naturally in the context of coupled light and matter; and to discuss some of the more interesting phenomena which may arise for matter coupled to light. Semiclassical be- haviour will be discussed in some sections, both because an understanding of semiclassical behaviour (i.e. classical radiation coupled to quantum me- chanical matter) is useful to motivate what phenomena might be expected; and also as comparison to the semiclassical case is important to see what new physics arises from quantised radiation. Contents Introduction Quantisation of electromagnetism Revision: Lagrangian for electromagnetism Eliminating redundant variables Canonical quantisation; photon modes Approximations of light-matter coupling Further reading Quantum electrodynamics in other gauges Freedom of choice of gauge and classical equations Transformation to the electric dipole gauge Electric dipole gauge for semiclassical problems Pitfalls of perturbation Further reading Jaynes Cummings model Semiclassical limit Single mode quantum model Many mode quantum model --- irreversible decay Further properties of collapse and revival Density matrices for 2 level systems Density matrix equation for relaxation of two-level system Dephasing in addition to relaxation Power broadening of absorption Further reading Resonance Fluorescence Spectrum of emission into a reservoir Quantum regression ``theorem'' Resonance fluorescence spectrum Further reading Quantum stochastic methods Quantum jump formalism Heisenberg-Langevin equations Fluctuation dissipation theorem Further reading Cavity Quantum Electrodynamics The Purcell effect in a 1D model cavity Weak to strong coupling via density matrices Examples of Cavity QED systems Further reading Superradiance Simple density matrix equation for collective emission Beyond the simple model Further reading The Dicke model Phase transitions, spontaneous superradiance No-go theorem: no vacuum instability Radiation in a box; restoring the phase transition Dynamic superradiance Further Reading Lasers and micromasers Density matrix equations for a micromaser and a laser Laser rate equations Laser Linewidth Further reading More on lasers Density matrix equation Spontaneous emission, noise, and parameter Single atom lasers Further reading Three levels, and coherent control Semiclassical introduction Coherent evolution alone; why does EIT occur Dark state polaritons Further reading Bibliography
