Laser photon merging in an electromagnetic field inhomogeneity

We study the effect of laser photon merging, or equivalently high harmonic generation, in the quantum vacuum subject to inhomogeneous electromagnetic fields. Such a process is facilitated by the effective nonlinear couplings arising from charged particle-antiparticle fluctuations in the quantum vacuum subject to strong electromagnetic fields. We derive explicit results for general kinematic and polarization configurations involving optical photons. Concentrating on merged photons in reflected channels which are preferable in experiments for reasons of noise suppression, we demonstrate that photon merging is typically dominated by the competing nonlinear process of quantum reflection, though appropriate polarization and signal filtering could specifically search for the merging process. As a byproduct, we devise a novel systematic expansion of the photon polarization tensor in plane wave fields.

Figure 1

Typical Feynman diagram contributing to the effect of quantum reflection [16]. For field strengths of the inhomogeneity well below the critical field strength (cf. main text), the leading contribution arises from diagrams with two couplings to the field inhomogeneity. As there is no energy transfer from static fields, the frequencies of the incident and outgoing photons match.

Figure 2

Cartoon of the photon merging process. In the presence of a stationary but spatially inhomogeneous electromagnetic field $2n$ laser photons of frequency $\omega$ can merge into a single photon of frequency $2n\omega$. Depending on the spatial field inhomogeneity, the propagation direction of the merged photons can differ from that of the incident probe photons. In curly braces we introduce our notation for the corresponding fields/polarizations and four-momenta; cf. also Eqs. (1), (26) and (35), as well as Fig. 3.

Figure 3

Schematic depiction of the two-photon merging process. Incident probe photons (wave-vector $\overrightarrow{\kappa }$, energy ${\kappa }^0=|\overrightarrow{\kappa }|=\omega$) hit a one-dimensional field inhomogeneity $\overrightarrow{B}(\mathrm{x})=B(\mathrm{x}){\overrightarrow{e}}_{\mathrm{z}}$ of width $w$ under an angle of $\theta$. Due to nonlinear effective couplings between electromagnetic fields mediated by virtual charged particle fluctuations, the field inhomogeneity can impact incident probe photons to merge and form an outgoing photon (wave-vector ${\overrightarrow{k}}_f$) of twice the energy of the incident probe photons, i.e., $k_f^0=|{\overrightarrow{k}}_f|=2\omega$. Most notably, the inhomogeneity can affect the outgoing merged photons to reverse their momentum component along ${\overrightarrow{e}}_{\mathrm{x}}$ with respect to the incident probe photons. The vectors ${\overrightarrow{a}}_1$, ${\overrightarrow{a}}_2$ and ${\overrightarrow{\epsilon }}^{(1)}$, ${\overrightarrow{\epsilon }}^{(2)}$ span the polarization degrees of freedom of the incident and outgoing photons, respectively. For the depiction we specialized to $\phi ={\phi }^{\prime }=0$ (cf. the main text).

Figure 4

Sketch of the envelope of a Gaussian beam intersecting the y–z plane in the vicinity of its waist under an angle of $\theta$ (cf. also Fig. 3). Given that the transverse cross-section area of the Gaussian beam at the beam waist is a circle of area $\sigma$, the intersection area is an ellipse with area $\frac{\sigma }{\cos \theta }$.

Figure 5

Choosing $\mathfrak{E}=B$ as an example, we depict the regimes where photon merging dominates quantum reflection and vice versa based on Eq. (57). Photon merging dominates quantum reflection in the regime in the lower left bounded by the blue (solid) and red (dotted) lines for Gauss and Lorentz type inhomogeneities, respectively.

Figure 6

Number of induced outgoing photons per shot ${\mathcal{N}}_{\text{merg}}$ due to the effects of laser photon merging and quantum reflection as a function of $\theta$, adopting the design parameters of the Jena high-intensity laser systems, JETI 200 and Polaris (cf. main text). The horizontal dashed line shows where the number of induced outgoing photons per shot becomes one. For quantum reflection this is the case for $\theta \ge 78^{\circ }$ [16]. Conversely, the number of outgoing merged photons reaches a maximum at $\theta \approx 87^{\circ }$ and stays below ${\mathcal{N}}_{\text{merg}}{|}_{\max }\approx 1.3\times {10}^{-4}$ throughout the interval $0\le \theta \le 90^{\circ }$; cf. Eq. (59) for the Gaussian inhomogeneity and the discussion below. For completeness, we note that ${\mathcal{N}}_{\text{Qref}}{|}_{\theta =0}\approx 3\times 10^{-29}$ while ${\mathcal{N}}_{\text{merg}}{|}_{\theta =0}\approx 3\times 10^{-228}$.