Relativistic calculation of the electron-momentum shift in tunneling ionization

We describe a procedure for the solution of the time-dependent Dirac equation. The procedure is based on the relativistic generalization of the matrix iteration method. We use this procedure to study electron-momentum distribution along the laser-beam propagation direction for the process of the tunneling ionization of a hydrogen atom. We found, in agreement with the experimental observations [C. T. L. Smeenk, L. Arissian, B. Zhou, A. Mysyrowicz, D. M. Villeneuve, A. Staudte, and P. B. Corkum, Phys. Rev. Lett. 106, 193002 (2011)], that relativistic effects lead to appreciable deviation of the distribution from the strict left-right symmetry present in the nonrelativistic case. The expectation value of the momentum along the laser-beam propagation direction grows linearly with intensity and follows closely the behavior of the expectation value of the kinetic energy divided by the speed of light. These features agree with the experimental results [C. T. L. Smeenk, L. Arissian, B. Zhou, A. Mysyrowicz, D. M. Villeneuve, A. Staudte, and P. B. Corkum, Phys. Rev. Lett. 106, 193002 (2011)].

Figure 1

Time dependence of the expectation values of operators $r$ and $\widehat{H}$. Evolution starts from the initial superposition of states with energies which differ by approximately $2c^2$.

Figure 2

Relativistic (solid red line) and nonrelativistic (dashed green line) energy spectra for the field intensity of $1\times {10}^{14}$ ${\mathrm{W}/\mathrm{cm}}^2$.

Figure 3

Distribution $W\left(p_x\right)$ for the field intensity of $5.05\times {10}^{14}$ ${\mathrm{W}/\mathrm{cm}}^2$ a.u.

Figure 4

Expectation value $\langle p_x\rangle$ (red crosses) as a function of the field intensity; $\frac{1}{c}\langle E_{\mathrm{kin}}\rangle$ (green dashed line).