Laser-dressed vacuum polarization in a Coulomb field

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 $r⪢1∕\ensuremath{\omega}$ from the nucleus ($\ensuremath{\omega}$ 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.