Laser-dressed vacuum polarization in a Coulomb field

A. I. Milstein, I. S. Terekhov, U. D. Jentschura, and C. H. Keitel

Phys. Rev. A 72, 052104 – Published 3 November 2005

Abstract

We investigate quantum electrodynamic effects under the influence of an external, time-dependent electromagnetic field, which mediates dynamic modifications of the radiative corrections. Specifically, we consider the quantum electrodynamic vacuum-polarization tensor under the influence of two external background fields: a strong laser field and a nuclear Coulomb field. We calculate the charge and current densities induced by a nuclear Coulomb field in the presence of a laser field. We find the corresponding induced scalar and vector potentials. The induced potential, in first-order perturbation theory, leads to a correction to atomic energy levels. The external laser field breaks the rotational symmetry of the system. Consequently, the induced charge density is not spherically symmetric, and the energy correction therefore leads to a “polarized Lamb shift.” In particular, the laser generates an additional potential with a quadrupole moment. The corresponding laser-dressed vacuum-polarization potential behaves like $1 ∕ r^{3}$ at large distances, unlike the Uehling potential, which vanishes exponentially for large $r$ . The energy corrections are of the same order of magnitude for hydrogenic levels, irrespective of the angular momentum quantum number. The induced current leads to a transition dipole moment which oscillates at the second harmonic of the laser frequency and is mediated by second-order harmonic generation in the vacuum-polarization loop. In the far field, at distances $r ⪢ 1 ∕ ω$ from the nucleus ( $ω$ is the laser frequency), the laser induces mutually perpendicular electric and magnetic fields, which give rise to an energy flux that corresponds to photon fusion leading to the generation of real photons, again at the second harmonic of the laser. Our investigation might be useful for other situations where quantum field theoretic phenomena are subjected to external fields of a rather involved structure.

Received 13 July 2005

DOI:https://doi.org/10.1103/PhysRevA.72.052104

Authors & Affiliations

A. I. Milstein and I. S. Terekhov

Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia

U. D. Jentschura and C. H. Keitel

Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

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Vol. 72, Iss. 5 — November 2005

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Images

Figure 1
Diagrams corresponding to the amplitude (11), including their multiplicities. The solid lines are the electron propagators, the dashed line with the cross denotes the Coulomb field, the wavy lines correspond to the laser photons, and the zigzag line is the induced field.Reuse & Permissions
Figure 2
(Color online) The familiar rotationally symmetric vacuum-polarization charge density around an atomic nucleus is severely distorted in a strong laser field. Here, we plot the additional laser-dressed, time-independent contribution to the renormalized charge density $δ ρ_{0}^{R} (\vec{r})$ , as induced by the laser field. The quantity $δ ρ_{0}^{R} (\vec{r})$ has to be added to the well-known expressions for the Uehling and Wichmann-Kroll parts (for a review see, e.g., [10]). The result for $δ ρ_{0}^{R} (\vec{r})$ is obtained after Fourier transformation of the momentum-space result given in Eq. (18) below, into coordinate space. Along the vertical axis, we have the propagation direction of the laser, which we assume to be linearly polarized ( $ξ_{1}^{2} \neq 0$ , $ξ_{2}^{2} = 0$ ). The polarization direction (laser electric field) is along the $x$ axis. We plot a characteristic surface given by a constant induced charge density—namely, the surface defined by the relation $δ ρ_{0} (\vec{r}) = (2 α ∕ 3 π) (Z e m^{3} ∕ π) {(E ∕ E_{0})}^{2} K$ , where $K$ is a numerical constant given by $K = \pm 1.204$ for the plot shown. Two areas with a positive induced charge density (red, aligned mainly along the $y$ axis, $K = + 1.204$ ) and a negative one are distinguished (the latter is in blue color and located along the propagation axis, $K = - 1.204$ ). As emphasized in the text, the total induced charge is zero.Reuse & Permissions
Figure 3
(Color online) Projections of the laser-induced part of the vacuum-polarization charge density looked upon from the $x$ , $y$ and $z$ axes, in (a), (b) and (c), respectively. The conventions are the same as in Fig. 2: the $z$ axis is the laser propagation direction, and the $x$ axis is the polarization axis (laser electric field). The picture provides a ray-traced impression of the charge density, represented as an absorptive medium. Red color is used for positive induced charge (areas located along the $x$ and $y$ axes); blue color is used for negative charge (areas located along the $z$ axis). The plots are as seen by an observer looking at the atomic nucleus from the specified directions. This is in contrast to Fig. 2, where the isosurface corresponding to a given, constant charge density is plotted. In the above graphical representations, the falloff of the induced charge density for large radial arguments can be distinguished clearly.Reuse & Permissions

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