Phase effects in laser-induced electron-positron pair creation

Probability rates of electron-positron pair creation in collisions of a relativistic nucleus with a two-color laser field are calculated using the $S$-matrix approach. The case when both components of the laser field have commensurate frequencies and comparable strengths is considered. Pronounced interference effects are observed in angular distributions of created particles as well as in the total probability rates of pair production. These interference effects show a significant phase dependence of the pair-creation process.

Figure 1

Angular maps of $R_5({\mathit{q}}_{\mathrm{f}},{\widehat{\mathit{p}}}_{{\mathrm{e}}^{-}})$ for the case when $\omega =m_{\mathrm{e}}{c}^2$ (in the chosen reference frame), $n_1=1$, $n_2=2$, and ${\mu }_1={\mu }_2=1$. The momentum transfer from the colliding proton is ${\mathit{q}}_{\mathrm{f}}=(0,0,-2m_{\mathrm{e}}c)$. Each panel corresponds to a different relative phase of both color fields. Here, the angles ${\theta }_{e^{-}}$ and ${\varphi }_{e^{-}}$ are the electron spherical angles measured in the coordinate frame such the $z$ axis is in the opposite direction to the laser field propagation direction, whereas the $x$ axis is along the polarization direction of both color fields.

Figure 2

Same as Fig. 1 but for the 183-photon pair-creation process, $R_{183}({\mathit{q}}_{\mathrm{f}},{\widehat{\mathit{p}}}_{{\mathrm{e}}^{-}})$, and for an even stronger laser field such that ${\mu }_1={\mu }_2=10$. For visual purposes, the rates $R_{183}({\mathit{q}}_{\mathrm{f}},{\widehat{\mathit{p}}}_{{\mathrm{e}}^{-}})$ have been raised to power 1/5.

Figure 3

Angular maps of $R_5({\mathit{q}}_{\mathrm{f}},{\widehat{\mathit{p}}}_{{\mathrm{e}}^{-}})$ (left panel) for the case when $\omega =m_{\mathrm{e}}{c}^2$, ${\mu }_1=\sqrt{2}$, and ${\mu }_2=0$. The right column shows $R_{183}({\mathit{q}}_{\mathrm{f}},{\widehat{\mathit{p}}}_{{\mathrm{e}}^{-}})$ for the same parameters except that ${\mu }_1=10\sqrt{2}$. In each case, the momentum transfer from the colliding proton is ${\mathit{q}}_{\mathrm{f}}=(0,0,-2m_{\mathrm{e}}c)$. For visual purposes, the rates $R_{183}({\mathit{q}}_{\mathrm{f}},{\widehat{\mathit{p}}}_{{\mathrm{e}}^{-}})$ have been raised to power 1/5.

Figure 4

Comparison of two shape functions $f_{\mathrm{train}}(k·x,\chi )$ given by Eq. (37) (red dashed line) and $f(k·x,\chi )$ described by Eq. (38) (blue solid line) for chosen values of the carrier-envelope phase $\chi$ [$\chi =0$ (upper panel) and $\chi =\pi /2$ (lower panel)].

Figure 5

Shape functions $f(k·x,\chi )$, $f_{\mathrm{U}}(k·x,\chi )$, and $f_{\mathrm{I}}(k·x,\chi )$ [defined by Eqs. (38), (51), and (58), respectively] for three chosen values of the carrier envelope phase $\chi$ [$\chi =0$ (blue solid line), $\chi =0.25\pi$ (red dashed line), and $\chi =0.5\pi$ (black dash-dotted line)]. The last panel shows the dependence of the normalization factors $N_{\mathrm{U}}\left(\chi \right)$ and $N_{\mathrm{I}}\left(\chi \right)$ on the phase $\chi$.

Figure 6

Modulus of the first and the third harmonic components for three shape functions considered in this paper; the solid red line corresponds to $f_{\mathrm{U},N}\left(\chi \right)$, the black dashed line is for $f_{\mathrm{I},N}\left(\chi \right)$, whereas the blue dash-dotted line is for $f_N\left(\chi \right)$.

Figure 7

Total probability rate of pair creation as a function of the phase $\chi$ for the case when the laser-field frequency is $\omega =m_{\mathrm{e}}{c}^2$ (in the chosen reference frame), whereas $\mu =1$ (upper-most panel), $\mu =0.1$ (middle panel), and $\mu =0.01$ (lower-most panel). In each panel, the results obtained for different shape functions discussed in this paper [i.e., $f(k·x,\chi )$ (diamonds), $f_{\mathrm{U}}(k·x,\chi )$ (circles), and $f_{\mathrm{I}}(k·x,\chi )$ (squares)] are presented.

Figure 8

Dependence of the ratio defined by Eq. (63) on the carrier-envelope phase $\chi$ for $\omega =m_{\mathrm{e}}{c}^2$ (in the chosen reference frame) and $\mu =1$. In the upper panel, we present the results for the case when the shape function is either normalized with respect to the ponderomotive energy (circles) or with respect to the intensity of the laser field (squares). In the lower panel, the results for the unnormalized shape function are shown (diamonds).