Vacuum birefringence in strong inhomogeneous electromagnetic fields

Birefringence is one of the fascinating properties of the vacuum of quantum electrodynamics (QED) in strong electromagnetic fields. The scattering of linearly polarized incident probe photons into a perpendicularly polarized mode provides a distinct signature of the optical activity of the quantum vacuum and thus offers an excellent opportunity for a precision test of nonlinear QED. Precision tests require accurate predictions and thus a theoretical framework that is capable of taking the detailed experimental geometry into account. We derive analytical solutions for vacuum birefringence which include the spatio-temporal field structure of a strong optical pump laser field and an x-ray probe. We show that the angular distribution of the scattered photons depends strongly on the interaction geometry and find that scattering of the perpendicularly polarized scattered photons out of the cone of the incident probe x-ray beam is the key to making the phenomenon experimentally accessible with the current generation of FEL/high-field laser facilities.

Figure 1

Sketch of the pump-probe type scenario intended to verify vacuum birefringence. A linearly polarized optical high-intensity laser pulse—wavevector $\mathit{\kappa }$, electric (magnetic) field $\mathbf{E}$ ($\mathbf{B}$)—propagates along the positive z axis. Its strong electromagnetic field couples to the charged particle-antiparticle fluctuations in the quantum vacuum, and thereby effectively modifies its properties to be probed by a counter-propagating x-ray beam (wavevector $\mathbf{k}$, polarization $\mathit{\epsilon }$). Vacuum birefringence manifests itself in an ellipticity of the outgoing x-ray photons (wavevector ${\mathbf{k}}^{\prime }$, polarization components along ${\mathit{\epsilon }}^{(1)}$ and ${\mathit{\epsilon }}^{(2)}$).

Figure 2

Pictogram of the three different cases (a)–(c) considered in this paper. The circle (orange) is the cross section of the pump beam at its waist, and the filled circle/ellipse (purple) is the cross section of the probe beam.

Figure 3

Plot of $\frac{T}{N}\frac{\mathrm{d}{N}^{(\perp )}}{\mathrm{d}{k}_{\mathrm{z}}^{\prime }}$ for case (a) as a function of $k_{\mathrm{z}}^{\prime }$. The induced photon signal is peaked at $k_{\mathrm{z}}^{\prime }=-\omega$ and rapidly falls off to zero. We depict results for different probe pulse durations $T$. The inlay shows a plot of the same quantity for $T\to \infty$ (${\phi }_0=0$) over a wider frequency range, adopting a logarithmic scale. Here, the strongly suppressed contributions to be associated with frequencies $\approx \omega \pm 2\mathrm{\Omega }$ are clearly visible.

Figure 4

Plot of $\frac{\mathrm{d}{N}^{(\perp )}}{\mathrm{d}\cos {\vartheta }^{\prime }}\equiv 2\pi {\int }_0^{\infty }\mathrm{d}{\omega }^{\prime }{\omega }^{\prime 2}\frac{{\mathrm{d}}^3{N}^{(\perp )}}{{\mathrm{d}}^3{k}^{\prime }}$ for case (b) and $\frac{\mathrm{d}{N}^{(\perp )}}{\mathrm{d}{\vartheta }^{\prime }}\equiv \int \mathrm{d}{k}_{\mathrm{z}}^{\prime }{\int }_0^{\infty }\mathrm{d}{\omega }^{\prime }{\omega }^{\prime }\frac{{\mathrm{d}}^3{N}^{(\perp )}}{{\mathrm{d}}^3{k}^{\prime }}$ for case (c) as a function of the emission angle ${\vartheta }^{\prime }$. For $T=30\mathrm{fs}$ the two curves basically fall on top of each other, and are well approximated by $n^{(\perp )}(\xi )=\exp \{-\xi (\frac{{\vartheta }^{\prime }}{\theta (w_0)}{)}^2\}$ with $\xi =1.13$. This profile is compatible with naive expectations: From $N^{(\perp )}\sim I^2$ and ${\omega }^{\prime }\approx \omega$ we might have guessed $\mathrm{d}{N}^{(\perp )}\sim n^{(\perp )}(\xi =1)$. For comparison, the intensity profile of a Gaussian probe beam of divergence $\theta (3w_0)$ is depicted in gray.