Fig. 3: The spreading time td of the photon field, angular spans Δθ∞ defining negative areas for td and the Mach cone of the emission.
Panel a We take ω = 0.51 eV, σ = 10−5 and the packet width σx = 1/σ ≈38 nm. The spreading time td (in picoseconds) from Eq. (9) is displayed for the electron velocity β = 0.65 and refractive index n = 1.46 (solid black), β = 0.7 and n = 1.46 (dashed blue), β = 0.9999 and n = 1.05 (dotted green) and β = 0.9999 and n = 1.3 (dot-dashed red line). The sharp maxima are nearby the angle \({\theta }_{{{{\rm{Ch.cl.}}}}}=\arccos 1/{u}_{p}n\) as illustrated by the arrows. The Cherenkov condition is not met for the black line, which is why the photon field quickly spreads during hundreds of attoseconds. Panel b The inverse spreading time is plotted for parameters n = 1.5, β = 0.9, σ = 10−4 m and ω = 51.1 eV (solid black), ω = 5.1 eV (dashed blue) and ω = 0.51 eV (dotted green line). In the angular span of a size Δθ∞ between two points given by Eq. (10) the spreading time turns negative due to the quantum recoil (ω/ε ≠ 0, see Eq. (11)). Panel c A close-up picture of the spreading time td for parameters n = 1.5, β = 0.999, ω = 5.1 eV, σ = 10−4 m, θCh.cl ≈48.1°. Panel d Formation of the Mach cone of the electron emission. The electron moves with velocity up exceeding the photon velocity uk. The photon field produces the Mach cone with an angle θMach according to Eq. (8).
