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Physics

Vacuum Self-Dressing of an Atom and Its Physical Effects

Roberto Passante, Lucia Rizzuto

Published Jun 6, 2025• 1 citation
Publisher

Abstract

We consider a multilevel atom such as an hydrogen atom, interacting with the quantum electromagnetic field, in the dressed ground state of the interacting system. We evaluate by perturbation theory the dressed ground state of the system, within dipole approximation; we investigate and review the effect of the self-dressing of the atom on several field and atomic observables. Specifically, we obtain general expressions of the renormalized electric and magnetic field fluctuations and energy densities around the atom, and analyze their scaling with the distance from the atom, obtaining approximated expressions in the so-called near and far zone. We also investigate nonlocal spatial field correlations around the atom. We stress how the quantities we evaluate can be probed through two- and three-body nonadditive Casimir-Polder dispersion interactions. We also investigate the effect of the self-dressing, i.e. of the virtual transitions occurring in the dressed ground state, on atomic observables, for example the average potential energy of the electron in the nuclear field; this also allows us to obtain a more fundamental quantum basis for the Welton interpretation of the Lamb shift of a ground-state hydrogen atom, in terms of the atomic self-dressing processes.

DOI
10.3390/physics7020020

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We consider a multilevel atom such as an hydrogen atom, interacting with the quantum electromagnetic field, in the dressed ground state of the interacting system. We evaluate by perturbation theory the dressed ground state of the system, within dipole approximation; we investigate and review the effect of the self-dressing of the atom on several field and atomic observables. Specifically, we obtain general expressions of the renormalized electric and magnetic field fluctuations and energy densities around the atom, and analyze their scaling with the distance from the atom, obtaining approximated expressions in the so-called near and far zone. We also investigate nonlocal spatial field correlations around the atom. We stress how the quantities we evaluate can be probed through two- and three-body nonadditive Casimir-Polder dispersion interactions. We also investigate the effect of the self-dressing, i.e. of the virtual transitions occurring in the dressed ground state, on atomic observables, for example the average potential energy of the electron in the nuclear field; this also allows us to obtain a more fundamental quantum basis for the Welton interpretation of the Lamb shift of a ground-state hydrogen atom, in terms of the atomic self-dressing processes.
An atom or a molecule is constituted by a set of bound electric charges with dynamics governed by the laws of quantum mechanics. These charges are sources of the quantized electromagnetic field which also binds them together, and thus the effects of their interaction with the field cannot be disregarded in principle. Consequently even overall neutral atoms, on which we focus our attention, are driven by a dynamics which is inextricably related to the dynamics of the quantized electromagnetic field. One of the most prominent aspects of the atom-held interaction is the existence of a cloud of virtual photons which dresses the atom even in the lowest possible energy state of the system. In this paper we review the static as well as the dynamic aspects of the theory of the virtual cloud around the neutral atoms. We begin by reviewing various forms of the atom-field coupling as well as various models of simplified atoms which will be used in the rest of the paper. The question then arises as how to characterize quantitatively the shape of the virtual cloud, and we show that the energy density of the electromagnetic field is a physical quantity suitable for this purpose. First a perturbative approach to calculating this shape is developed and applied to several physical models of a ground-state dressed atom. The next step is to consider virtual clouds which are out of equilibrium and examine their time development. This leads to the concept of half-dressed sources, which are discussed in a different physical context both in the absence and in the presence of real photons. In particular the role of the virtual cloud in ensuring causality of the field propagating in dressing and undressing processes is emphasized. Finally, the nature of the virtual cloud is further discussed in the light of theories concerning the dynamics of an atomic pair, the quantum theory of measurement and the effects of a driving electromagnetic field.
We consider a system formed by a two-level atom initially in its bare ground state and the electromagnetic field in the absence of radiation. The interaction of the atom with the radiation field leads to the formation of a dressed ground state. To describe the dynamics of self-dressing, we study the time evolution of the permanence amplitude in the initial state. We adopt non-relativistic QED and take the atom-field interaction Hamiltonian in the minimal coupling form, within the electric dipole approximation. In order to follow the evolution of the permanence amplitude for long enough time periods we use the van Hove resolvent technique with appropriate partial summation of Feynman graphs. We show that the self-dressing process presents the typical behaviour of a decay process and occurs with a timescale a *-1 , which is long compared with both the atom's inverse transition frequency 0 -1 and the excited state decay time -1 . This time is comparable with the lifetime of the excited state due to ordinary two-photon decay. We consider the reasons for such a long time period of self-dressing and make comparison with previous results.
Introduction . The main theme of this chapter is the explicit calculation of the shape of the virtual cloud surrounding different kinds of ground-state sources. In Section 7.1 two of the three examples considered in Section 6.1 are taken up again, and it is argued that a convenient description of the shape of the virtual cloud is given by the energy density of the field around the source. This energy density is evaluated in detail for the static source of mesons and for the Fröhlich polaron. Section 7.2 is dedicated to an analogous calculation of the electric energy density around a two-level atom within a perturbation scheme. The virtual cloud around a two-level atom is again the subject of Section 7.3, where we evaluate the magnetic energy density as well as the coarse-grained energy density. From the results of the first three sections we conclude that the qualitative picture of the virtual cloud proposed in Section 6.2 is well founded. Moreover, the discussion of the two-level atom leads us to an important conclusion: the space surrounding an atom can be divided into a near zone and a far zone, where the energy density of the virtual cloud behaves differently as a function of the distance from the atom. In Section 7.4 we evaluate the energy density of the virtual cloud surrounding a slow free electron, separating classical and quantum contributions.
We consider the Casimir-Polder interaction between two atoms, one in the ground state and the other in its excited state. The interaction is time dependent for this system, because of the dynamical self-dressing and the spontaneous decay of the excited atom. We calculate the dynamical Casimir-Polder potential between the two atoms using an effective Hamiltonian approach. The results obtained and their physical meaning are discussed and compared with previous results based on a time-independent approach, which uses a nonnormalizable dressed state for the excited atom.