Modified light-cone condition via vacuum polarization in a time-dependent field

The onset of unconventional vacuum properties in intense fields has long been an active field of research. In this paper the vacuum polarization effect is investigated via a pump probe scheme in which a probe light propagates in the vacuum excited by a standing wave composed of two counterpropagating laser beams. The modified light-cone condition of the probe light is derived analytically for the situation that it passes through the electric (magnetic) antinode plane of the pump field and thus the probe light effectively experiences a time-dependent electric (magnetic) field. The derivation does not follow the commonly adopted assumption of treating the pump field as a constant field. Differences from the conventional light-cone conditions are identified. The light-cone condition is used to calculate the ellipticity value for a conceptual vacuum birefringence measurement. The quasistatic treatment is partly justified and the implications of the unconventional correction to previous results are discussed.

Figure 1

Scheme of the conceptual experimental setup. (a) A standing-wave pump field is formed by two counterpropagating linearly polarized laser waves with wave vectors $\pm {\vec{k}}_{\varphi }$ along the $z$ axis. The field has an electric component in the ${\widehat{e}}_x$ direction and a magnetic component in the ${\widehat{e}}_y$ direction. A probe beam with wave vector $\vec{k}$ propagates on an electric antinode plane where $z$ is a fixed value. The situation for the beam propagating on a magnetic antinode plane is similar. Here the probe beam is drawn to be along the $y$ axis for viewing convenience. (b) Two-dimensional display of the experimental setup on the $x\text{−}y$ plane. As the probe beam passes through the electric antinode plane, it experiences a spatially uniform but time-varying electric field $\vec{E}=E_x{\widehat{e}}_x=\mathcal{E}{\widehat{e}}_{x}cos{\omega }_{\varphi }t$. Here ${\theta }_E$ denotes the angle between the directions of $\vec{k}$ and the electric pump field $\vec{E}$.