On the Superradiant Phase in Field-Matter Interactions

TL;DR

This paper demonstrates that symmetry-adapted semi-classical states closely approximate the quantum ground and first excited states of the Dicke model, providing analytical expressions and clarifying the behavior of observables near the phase transition.

Contribution

It introduces symmetry-adapted semi-classical states as accurate approximations for the Dicke model's low-energy states, with analytical forms and improved understanding of phase transition behavior.

Findings

High overlap of semi-classical states with exact quantum states except near the phase transition

Analytical expressions for expectation values of observables

Photon and atom number expectations show no singularities at the phase transition

Abstract

We show that semi-classical states adapted to the symmetry of the Hamiltonian are an excellent approximation to the exact quantum solution of the ground and first excited states of the Dicke model. Their overlap to the exact quantum states is very close to 1 except in a close vicinity of the quantum phase transition. Furthermore, they have analytic forms in terms of the model parameters and allow us to calculate analytically the expectation values of field and matter observables. Some of these differ considerably from results obtained via the standard coherent states, and by means of Holstein-Primakoff series expansion of the Dicke Hamiltonian. Comparison with exact solutions obtained numerically support our results. In particular, it is shown that the expectation values of the number of photons and of the number of excited atoms have no singularities at the phase transition. We comment on why other authors have previously found otherwise.