Probing vacuum birefringence using x-ray free electron and optical high-intensity lasers

Felix Karbstein and Chantal Sundqvist

Phys. Rev. D 94, 013004 – Published 6 July 2016

Abstract

Vacuum birefringence is one of the most striking predictions of strong field quantum electrodynamics: Probe photons traversing a strong field region can indirectly sense the applied “pump” electromagnetic field via quantum fluctuations of virtual charged particles which couple to both pump and probe fields. This coupling is sensitive to the field alignment and can effectively result in two different indices of refraction for the probe photon polarization modes giving rise to a birefringence phenomenon. In this article, we perform a dedicated theoretical analysis of the proposed discovery experiment of vacuum birefringence at an x-ray free electron laser/optical high-intensity laser facility. Describing both pump and probe laser pulses realistically in terms of their macroscopic electromagnetic fields, we go beyond previous analyses by accounting for various effects not considered before in this context. Our study facilitates stringent quantitative predictions and optimizations of the signal in an actual experiment.

Received 1 June 2016

DOI:https://doi.org/10.1103/PhysRevD.94.013004

Physics Subject Headings (PhySH)

Laser applications Quantum electrodynamics

Lasers

Particles & Fields

Authors & Affiliations

Felix Karbstein^1,2,* and Chantal Sundqvist^1,2,†

¹Helmholtz-Institut Jena, Fröbelstieg 3, 07743 Jena, Germany
²Theoretisch-Physikalisches Institut, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany

^*felix.karbstein@uni-jena.de
^†chantal.sundqvist@uni-jena.de

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Vol. 94, Iss. 1 — 1 July 2016

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Images

Figure 1
Feynman diagram of the leading-order process evaluated in Eq. (3), inducing signal photons of wave vector $k^{'}$ and polarization $p^{'}$ . The process is cubic in the combined macroscopic electromagnetic fields of the pump and probe pulses represented by wiggly lines ending at crosses. The dominant process features a single coupling to the x-ray probe ( $\times$ ) and a quadratic coupling to the high-intensity pump ( $\otimes$ ) fields.
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Figure 2
Sketch of the scenario considered here. The transverse profiles of the pump (beam axis coincides with the $z$ axis) and probe laser (beam axis along $\hat{k}$ ) beams are depicted in orange and blue, respectively. The pump is modeled as a linearly polarized, pulsed Gaussian beam of waist $w_{0}$ at $z = 0$ . The normalized electric ${\hat{e}}_{E}$ ( ${\hat{e}}_{e}$ ) and magnetic ${\hat{e}}_{B}$ ( ${\hat{e}}_{b}$ ) field vectors of the pump (probe) are perpendicular to its propagation direction ${\hat{e}}_{z}$ ( $\hat{k}$ ). Their orientation is controlled by an angle parameterizing rotations around each beam’s propagation axis. As the x-ray probe is substantially less focused than the pump, in the interaction volume we model it as of constant width; in the far-field the divergence of the probe needs to be accounted for (cf. main text). We allow for generic elliptically shaped probe cross-sections: To this end we introduce two different probe beam waists ${w_{1}, w_{2}}$ associated with two perpendicular transverse directions resembling ${{\hat{e}}_{e}, {\hat{e}}_{b}}$ , but being parameterized by an independent angle. The vector $x_{0}$ allows for describing a finite impact or spatial displacement of the foci. We look for signal photons scattered into ${\hat{k}}^{'}$ direction in the far-field. The associated transverse polarization vectors are ${ε_{(1)}, ε_{(2)}}$ .
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Figure 3
Plot of the ratio $\frac{d^{2} N_{⊥}}{d φ^{'} d \cos ϑ^{'}} / (\frac{d^{2} N}{d φ^{'} d \cos ϑ^{'}} P)$ as a function of the polar angle $ϑ^{'}$ for a probe of asymmetric cross section; waist parameters $\frac{w_{1}}{w_{0}} = 3$ for $φ^{'} = δ_{0}$ and $\frac{w_{2}}{w_{0}} = \frac{1}{10}$ for $φ^{'} = δ_{0} + \frac{π}{2}$ . The blue (solid) curve for $φ^{'} = δ_{0}$ becomes unity at $ϑ^{'} = σ_{1} = 23.12 μ rad$ . In the depicted angle range, the red (dashed) curve for $φ^{'} = δ_{0} + \frac{π}{2}$ looks consistent with a straight line. In fact, it slowly bends upward to surpass unity for $ϑ^{'} = σ_{2} = 5129.38 μ rad$ (cf. also Table 1). The two curves depicted here are continuously related as a function of the azimuthal angle $φ^{'}$ (cf. the main text).
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Figure 4
Differential number of perpendicularly polarized signal photons $\frac{d^{2} N_{⊥}}{d φ^{'} d \cos ϑ^{'}}$ plotted as a function of $ϑ^{'}$ for a probe of asymmetric cross-section; waist parameters $\frac{w_{1}}{w_{0}} = 3$ for $φ^{'} = δ_{0}$ , and $\frac{w_{2}}{w_{0}} = \frac{1}{10}$ for $φ^{'} = δ_{0} + \frac{π}{2}$ . The segment of the blue (solid) curve highlighted in gray fulfills $ϑ^{'} \geq σ_{1}$ , i.e., corresponds to signal photons emitted at an azimuthal angle of $φ^{'} = δ_{0}$ which could be detected employing state-of-the-art polarization purity measurements. An analogous regime also exits for the red (dashed) curve. However, it is only of academic relevance as it lies far outside the depicted angle interval beyond $ϑ^{'} = σ_{2} = 5129.38 μ rad$ (cf. Table 1), where $\frac{d^{2} N_{⊥}}{d φ^{'} d \cos ϑ^{'}}$ has essentially dropped to zero. Correspondingly, no signal photons are to be detected in $φ^{'} = δ_{0} + \frac{π}{2}$ direction. The two curves depicted here are continuously related by means of the angle $φ^{'}$ (cf. main text).
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