Adiabatic invariants for the regular region of the Dicke model

TL;DR

This paper introduces adiabatic invariants as approximate second integrals of motion in the regular energy region of the non-integrable Dicke model, using Born-Oppenheimer approximations to explain observed smooth variations in observables.

Contribution

It develops a theoretical framework employing adiabatic invariants via Born-Oppenheimer approximation for the Dicke model's regular energy region, validated against numerical results.

Findings

Adiabatic invariants approximate second integrals of motion in the Dicke model.

Theoretical predictions agree well with numerical results.

Framework explains smooth observable variations in the regular energy regime.

Abstract

Adiabatic invariants are introduced and shown to provide an approximate second integral of motion for the non-integrable Dicke model, in the energy region where the system exhibits a regular dynamics. This low-energy region is always present and has been described both in a semiclassical and a full quantum analysis. Its Peres lattices exhibit that many observables vary smoothly with energy, along lines which beg for a formal description. It is shown how the adiabatic invariants provide a rationale to their presence in many cases. They are built employing the Born-Oppenheimer approximation, valid when a fast system is coupled to a much slower one. As the Dicke model has a one bosonic and one fermionic degree of freedom, two versions of the approximation are used, depending on which one is the faster. In both cases a noticeably accord with exact numerical results is obtained. The employment of the adiabatic invariants provides a simple and clear theoretical framework to study the physical phenomenology associated to this energy regime, far beyond the energies where the quadratic approximation can be employed.