Interference effects in Bethe-Heitler pair creation in a bichromatic laser field

We study the creation of electron-positron pairs in the superposition of a nuclear Coulomb field and a two-color laser field of high intensity. Our focus lies on quantum interference effects, which may arise if the two laser frequencies are commensurable. We show that the interference manifests in the angular distributions of the created particles, which are discussed in the nuclear rest frame and the laboratory frame. Additionally, we demonstrate that the total pair-production rates can be affected by interference and identify the relative phase between the two laser modes, which optimizes the pair-production yield.

Figure 1

The applied pair creation scheme: two linearly polarized plane waves with perpendicular field vectors shined along the $z$ axis onto a counterpropagating nucleus, creating an electron-positron pair.

Figure 2

Angular-differential partial rates for laser pairs $(1,2)$ and $(1,3)$ with total photon energy $1.1\mathrm{MeV}$ in the nuclear rest frame: (a) laser pair $(1,2)$ with ${\xi }_1=1.06\times {10}^{-2}$ and ${\xi }_2=0.1$ and (b) laser pair $(1,3)$ with ${\xi }_1=3.59\times {10}^{-4}$ and ${\xi }_2=0.1$. The frequencies are chosen such that the total photon energy is reached for ${\tilde{n}}_1$ photons from the first mode and ${\tilde{n}}_2$ photons from the second mode. The emission angle $\theta$ is measured with respect to the laser propagation direction. The relative phase is $\phi =0$. Neither laser pair shows significant contribution from an interference term.

Figure 3

Angular-differential partial rates for (a) laser pair $(2,4)$ with ${\xi }_1=4.62\times {10}^{-5}$ and ${\xi }_2=0.01$ and (b) laser pair $(2,6)$ with ${\xi }_1=2.48\times {10}^{-7}$ and ${\xi }_2=0.01$. These two laser pairs show a large contribution from interference terms. For the relative phase $\phi$ set to zero, interference is destructive for $(2,4)$ and constructive for $(2,6)$.

Figure 4

Variation of the total photon energy of the angular-differential partial rates for laser pair $(2,4)$: (a) total photon energy of $1.25\mathrm{MeV}$, ${\xi }_1=5.07\times {10}^{-5}$, and ${\xi }_2=0.01$ and (b) total photon energy of $1.35\mathrm{MeV}$, ${\xi }_1=5.38\times {10}^{-5}$, and ${\xi }_2=0.01$. As the total photon energy increases, the relative contribution from interference decreases. Additionally, the shape of the interference terms changes, leading to a zero crossing in Fig. 4b. The intensity parameter ${\xi }_1$ has been adjusted to maintain equally contributing direct terms.

Figure 5

Individual terms and several subsums of the angular-differential partial rates of laser pair $(2,4)$ for two photon energies: (a) 1.1 MeV and (b) 1.35 MeV. Here $\sum$ all is the sum over all individual terms, $\sum$ direct is the sum of the two terms stemming solely from the first or the second laser mode, $\sum$ phase independent is the latter plus the symmetrically mixed term, and $\sum$ interference is the sum of the two interference terms. The relative phase is $\phi =\frac{\pi }{2}$. Otherwise, the parameters are the same as in Figs. 3a and 4b.

Figure 6

Phase variation in the sums of all terms (top panels) and the interference contributions (bottom panels) of the angular-differential partial rates of laser pair $(2,4)$ for two energies: (a) 1.1 MeV and (b) 1.35 MeV. The sum of the direct terms as in Fig. 5 is given for reference. For laser pair $(2,4)$ the interference contribution is maximal (maximally constructive) for the relative phase $\phi =\frac{\pi }{2}$, is minimal (maximally destructive) for $\phi =0$, and vanishes for $\phi =\frac{\pi }{4}$. The other parameters are the same as in Figs. 3a and 4b, respectively.

Figure 7

Comparison of (top) the scaled phase variation in the total pair-production rates and (bottom) the measure defined by the absolute maximum of the sum of the normalized electric fields from Eq. (43) for several laser pair combinations. Line types and colors are identical for both panels. As before $\phi ={\phi }_1$ and ${\phi }_2=0$. To achieve better comparability the total rates have been scaled by a factor of $1.45\times {10}^9$ for laser pair $(2,6)$ and by $1.35\times {10}^{18}$ for $(2,8)$.

Figure 8

Individual terms and several subsums of the angular-differential partial rates of laser pair $(2,4)$ for two photon energies transformed with a Lorentz factor $\gamma$ of 50 to the laboratory frame: (a) total photon energy (in the nuclear rest frame) of $1.1\mathrm{MeV}$ and (b) total photon energy (in the nuclear rest frame) of $1.35\mathrm{MeV}$. The notation, legend, and parameters are identical to those in Fig. 5.

Figure 9

Phase variation in the sums of all terms (top panels) and the interference contributions (bottom panels) of the angular-differential partial rates of laser pair $(2,4)$ for two energies transformed to the laboratory frame ($\gamma =50$): (a) total photon energy (in the nuclear rest frame) of $1.1\mathrm{MeV}$and (b) total photon energy (in the nuclear rest frame) of $1.35\mathrm{MeV}$. The sum of the direct terms as in Fig. 8 is shown for reference. The notation, legend, and parameters are identical to those in Fig. 6.