Mathematical Methods in Quantum Optics: the Dicke Model

TL;DR

This paper explores mathematical techniques like catastrophe formalism and group theory to analyze the phase transition in the Dicke model of quantum optics, providing universal curves and critical exponents for finite atom numbers.

Contribution

It introduces multiple mathematical approaches to study the Dicke model's phase transition, deriving universal parametric curves and the critical coupling's dependence on atom number.

Findings

Universal parametric curves for quadrature and photon number expectations

Critical atom-field coupling as a function of atom number

Critical exponent derived for the phase transition

Abstract

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.