Abstract
In this paper, we consider some second-order effective Hamiltonians describing the interaction of the quantum electromagnetic field with atoms or molecules in the nonrelativistic limit. Our procedure is valid only for off-energy-shell processes, specifically virtual processes such as those relevant for ground-state energy shifts and dispersion van der Waals and Casimir-Polder interactions, while on-energy-shell processes are excluded. These effective Hamiltonians allow for a considerable simplification of the calculation of radiative energy shifts, dispersion, and Casimir-Polder interactions, including in the presence of boundary conditions. They can also provide clear physical insights into the processes involved. We clarify that the form of the effective Hamiltonian depends on the field states considered, and consequently different expressions can be obtained, each of them with a well-defined range of validity and possible applications. We also apply our results to some specific cases, mainly the Lamb shift, the Casimir-Polder atom-surface interaction, and the dispersion interactions between atoms, molecules, or, in general, polarizable bodies.
Highlights
That is, the quantum theory of atoms and molecules interacting with the electromagnetic field in the nonrelativistic limit, several processes of great interest are of a high order in atom–field coupling [1,2,3]
All these possibilities have fostered the investigation of effective Hamiltonians containing an interaction term that is at least quadratic in the atom–field coupling, where the response of the atom is included in quantities such as, for example, its polarizability, allowing for a considerable reduction of the perturbative order required for calculating specific processes and of the number of relevant Feynman diagrams [6]
Resummation techniques, in which the polarizability is summed to any order, have been developed, and they could be of great importance for the evaluation of radiative processes such as the Lamb shift and van der Waals interactions for nanostructured materials [16,17,18]
Summary
That is, the quantum theory of atoms and molecules interacting with the electromagnetic field in the nonrelativistic limit, several processes of great interest are of a high order in atom–field coupling [1,2,3]. The multipolar-coupling Hamiltonian (1) contains a term equal to 2π d3r(P⊥(r)), where P⊥(r) is the transverse part of the polarization field P(r) = ∑i erδ(r − ri), with ri being the position of an atomic electron [1,19,20] This is a second-order term in the electric charge that, it is important for the Lamb shift, does not contribute to the dispersion interactions [1,6,20,21], and for this reason we do not include it, except whenever necessary. On the basis of the results obtained we will find the relative effective Hamiltonians in the various cases, according to the relevant photon states involved, and point out the range and limit of application of the specific forms obtained
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