Effective Hamiltonians in Nonrelativistic Quantum Electrodynamics

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.

Dynamical Casimir effects with atoms: From the emission of photon pairs to geometric phases

“…Microscopic dynamical Casimir effect. -Within the electric dipole approximation, the atom-field interaction is described by the Hamiltonian (for alternative models and a more general discussion see [48][49][50])…”

Nonlocal Static and Dynamical Vacuum Field Correlations and Casimir–Polder Interactions

“…showing also in this case some nonlocal aspect of the resonant term of the interaction energy. Finally, we wish to mention that similar considerations about the nonlocal features of the three-body component of the dynamical dispersion interaction between three atoms can be obtained by evaluating them exploring effective Hamiltonians [40,49], as discussed in detail in [13,24,51] also with reference to the measurement of the three-body component of dispersion interactions which involves the overall system and thus it is intrinsically nonlocal.…”

“…In the two points where atoms B and C are located, fluctuations of the electric and magnetic fields exist, and, as shown in the previous sections, these field fluctuations are spatially correlated; thus, for each field mode, they induce and correlate dipole moments in the two atoms. We have, in the spirit of a linear response theory [9,11,12,20,40],…”

Time-Dependent Effective Hamiltonians for Light–Matter Interactions

“…Another important aspect is that the effective Hamiltonian ( 11 ) is convenient only when first-order perturbation theory in Hamiltonian ( 3 ) vanishes, even though regularization techniques may render it applicable if this is not the case [ 8 ]. We may separate the main applications of the effective Hamiltonian into two groups: (i) the molecule remains in the same internal state during the entire process, and the expectation value of the electric dipole operator in this state is zero; (ii) the molecule undergoes a transition between two internal states, but the electric dipole operator is unable to connect these two states.…”

“…Here comes the convenience of working with effective Hamiltonians, which are tailored for each specific application, bringing several physical insights and shortening the technical calculation to a much simpler and lower perturbative order analysis. An insightful example is the dynamical polarizability (DP) Hamiltonian, obtained by R. Passante and collaborators [ 7 , 8 ], which is built directly on the molecular dynamical polarizability instead of its electric dipole operator, capturing better the physics governing the interaction. Indeed, nonpolar molecules do not possess permanent electric dipole moments, and their interaction is possible only due to virtual internal transitions that are automatically taken into account by the dynamical polarizability.…”