Overview of the superradiant phase transition: the Dicke model

The real interaction between matter and electromagnetic radiation is too complicated for a complete theoretical investigation. Nevertheless, in some cases the problem admits an amazing simplification which allows one to consider interesting phenomena in the framework of rather simple models having even exact solutions. A model, which describes in the dipolar approximation the interaction of N two-level atoms with a quantized radiation field in an ideal cavity with volume V, bears the name of Dicke. This model is of key importance for describing dynamical, collective and coherent effects in quantum optics. Since 1974, when Hepp and Lieb rigorously proved that the Dicke model exhibits a second order phase transition from the normal to a superradiant phase, its thermodynamic properties have been studied in detail in the context of critical phenomena and solid state physics. Quite recently, a new aspect emerged when it was realized that the quantum phase transition of the model is relevant to quantum information and quantum computing. Various physical approximations have been extensively debated in the above mentioned fields of research. Here, an attempt is made to review in a rigorous manner the thermodynamic properties of the original Dicke model and its different generalizations. Some new results concerning relations between different indicators of criticality are presented as well.