Rabi hamiltonian and geometric phases - Calderón Krejci, Juan Enrique (2016)
This thesis addresses geometric phases that appear when a two-level atom interacts with a quantized one-mode electromagnetic field, a model that is described by the Rabi Hamiltonian (RH). As it is known, the RH has no closed-form solution; nevertheless, when the coupling between the atom and field is weak, the rotatingwave approximation (RWA) can be applied. This results in the Jaynes-Cummings Hamiltonian (JCH), which is a useful analytically solvable approximation of the former one. Whenever the RWA can be applied, physical phenomena predicted within the Rabi model should also show up within the Jaynes-Cummings model; otherwise, the approximation would be physically inconsistent. This issue became controversial after a recent claim concerning Berry phases in the RH. According to this claim, the RWA breaks down at all values of the coupling between the atom and field. The results of this research, numerical calculations of Berry phases in the RH, showed that this is not the case and that claims on the contrary are inconsistent with an analytical argument regarding the Rabi model. Additionally, these results also converge to the corresponding ones obtained with the JCH, concluding that the RWA consistently applies when dealing with Berry phases, as expected. Finally, it is argued that the appearance of Berry phases does not depend on adiabatic conditions, hence the appropriate framework is the kinematic one, which contains Berry’s phase as a special case of the geometric phase. It is also argued that the Hamiltonian does not play an essential role in the whole, except as a provider of the eigenvectors used in the calculation of geometric phases. This brings to the fore the essential feature on which the geometric phase depends, which is the geometry of the ray space. This space depends on the types of evolutions being considered. This point is illustrated by addressing a different unitary transformation in the Schwinger model