LWP:Absence of vacuum induced Berry phases without the rotating wave approximation in cavity QED

by Jonas Larson
Physics Review Letters 108, 033601 (2012)

Some ten years ago, Berry phase was studied in the Jaynes-Cummings model. In this paper, Larson focuses on proving that such a geometric phase arises from the use of the rotating wave approximation.

First, Larson reviews the results fro the Jaynes-Cummings model, that is, under the rotating wave approximation, and derives the accumulated Berry phases by exact diagonalization of the Hamiltonian. Then, he presents a semiclassical approximation derivation by using canonical operators for the field and Born-Oppenheimer approximation to obtain a pair of adiabatic potentials that lead to semiclassical conical energy surfaces giving a Berry phase which is identical to that given by the exact calculation for large photon number. 

Once the validity of this semiclassical approximation is provided by this example, Larson uses the same method to find the semiclassical energy surfaces for the case without the rotating wave approximation, that is, Rabi Hamiltonian, and shows that these surfaces lack the conical intersection that gives a non zero Berry phase. Actually, the semiclassical energy surfaces for this case intersect along the line given by x=0 instead of just a point. It is noted that the author carried a numerical diagonalization of the Hamiltonian and the results were confirmed.

Furthermore, Larson revisits a lambda-atom configuration where a Raman coupling scheme is used and shows that by assuming far detuned driving fields such that adiabatic elimination can be carried on, without any rotating wave approximation, the semiclassical energy surfaces for the system do not intersect at a single point and Berry phase must be zero all the time.

He also discusses the case of a single atom in a bichromatic cavity and shows that within the used semiclassical formalism there are no vacuum induced Berry phases. A trio of other systems are presented with the same result: no Berry phase when the rotating wave approximation is not used.    

It concludes, that the presence of a Berry phase in the Rabi model and other models discussed (adiabatic elimination, bichromatic cavities, etc) in the paper is a result of introducing the rotating wave approximation and not inherent to the models. Thus, a flag is raised: whenever Berry phase is the subject of study one has to be mindful of the approximations used independently of the coupling regime at hand. It may be even possible that this result may change if a multi-level approximation is taken instead of the typical two-level restriction of the Rabi model. It also points the interest of studying vacuum induced Berry phase in nonadiabatic processes. 

This is a very nice paper to read, it presents a concrete, straight to the fact introduction of geometrical phases in cavity-QED and the results are presented in a clear manner. If you are a fan of the Jaynes-Cummings model, you cannot leave this article unread.