Fixed bound atom interacting with a coherent pulse of quantized radiation: Excitation and short-time behavior

Abstract

A model of a coupled atom---quantum-field system is presented, consisting of a time-independent Hamiltonian and a realistic dynamical initial condition. The Hamiltonian describes a fixed bound atom coupled to the complete electromagnetic field. The initial condition is a consistent quantum-mechanical description of an atom in the exact ground state of the coupled atom-field system, on which is superimposed an approaching quasiclassical excitation of the field which will cause the interesting dynamical processes later on. Tools are presented for the manipulation of this model, with emphasis towards series expansions of all relevant quantities suitable for small strength $\ensuremath{\lambda}$ of the atom-field coupling and (potentially) large magnitude $\ensuremath{\alpha}$ of the coherent field excitation, such that the "influence of the coherent excitation on the atom" (i.e., the product $\ensuremath{\lambda}\ensuremath{\alpha}$) remains constant as $\ensuremath{\lambda}$ tends to zero. The following results are obtained, at the lowest nontrivial order in $\ensuremath{\lambda}$, in an approximation suitable for times much shorter than any radiative lifetime of the atom: (a) The change in free-field energy due to the interaction with the atom is exactly compensated at all times by the change in bare-atom energy plus the atom-field coupling energy. This checks the consistency of the proposed scheme. (b) The frequency distribution of this change in free-field energy has a (somewhat unexpected) component, with a smooth spectrum, spreading over the whole width of the spectrum of the exciting pulse, in addition to $\ensuremath{\delta}$ and dispersive singularities centered on the atomic transition frequencies. This frequency distribution is discussed in detail and its features are illustrated by a number of figures, in a simple case, numerically tractable. (c) The heterodyne detection of the radiation emitted by the atom is investigated, with emphasis on the issue of causality and propagation of light. Strict causality is obtained, without further approximation, in a model with an atom-field coupling involving the field at a single point and a detector similarly sensitive to properties of the field at a single point. This result breaks down if one uses nonlocal approximations such as the rotating-wave approximation or a detection observable ${\mathbf{E}}^{\ensuremath{-}}(\mathbf{r})\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{E}}^{+}(\mathbf{r})$. Finally, an appendix is devoted to the use, in quantum mechanics, of bases moving with respect to each other.