Mathematical Methods in Quantum Optics: the Dicke Model

We show how various mathematical formalisms, specifically the catastrophe formalism and group theory, aid in the study of relevant systems in quantum optics. We describe the phase transition of the Dicke model for a finite number N of atoms, via 3 different methods, which lead to universal parametric curves for the expectation value of the first quadrature of the electromagnetic field and the expectation value of the number operator, as functions of the atomic relative population. These are valid for all values of the matter-field coupling parameter, and valid for both the ground and first-excited states. Using these mathematical tools, the critical value of the atom-field coupling parameter is found as a function of the number of atoms, from which its critical exponent is derived.