Laser-induced level shifts and splittings in multiphoton pair creation

Using numerical solutions to the Dirac equation, we examine the role of energy shifts on the energy spectrum of created positrons in the laser-induced electron-positron pair-creation process in the presence of a highly charged model nucleus. We suggest that at those laser intensities, where the created electrons can be captured by the nucleus, the inclusion of the field-induced energy level shifts is crucial to predict correct positron energy spectra, especially close to the threshold for this process with regard to the laser frequency. For the interesting case where the laser frequency is tuned to the bound state energy differences, the coherence associated with possible electronic Rabi oscillations is transferred to the generated positrons, leading to new split peaks in their energy spectrum.

Figure 1

Sketch of the relevant bare energy scales, the shape of the binding potential and the three bound states, which can become populated during the pair-creation process.

Figure 2

(a) The time dependence of the energy $e(t)\equiv ⟨\alpha (t)|H(r,t)|\alpha (t)⟩$ for an electron that is initially in the ground state $|\alpha ⟩$ for the potential used in our calculations (with $V_0=1.8m{c}^2$, $w=0.3\lambda$, $D=3.428\lambda$, $\eta \omega =0.48m{c}^2$ and $\xi =0.5$). (b) For comparison, we also show $e(t)$ for a much weaker bound and nonrelativistic ground state with energy $E_{\alpha }=0.716m{c}^2$, obtained for a different binding potential with $V_0=0.47m{c}^2$, $w=0.1\lambda$, $D=3\lambda$, $\hslash \omega =1.0m{c}^2$ and $\xi =0.5$). The dashed and dotted lines are the contributions to $e(t)$ from $⟨H_0⟩$ and $-q$$⟨\mathit{\alpha }\mathbf{A}⟩$, respectively.

Figure 3

The three dressed bound state energies $E(\xi ,\omega )$ as a function of the scaled field strength $\xi$ for the laser frequencies $\hslash \omega /(m{c}^2)=0.45$ and 0.55. For comparison, we have also included the threshold shift $E_{p=0}(\xi )=-m{(1+{\xi }^2/2)}^{1/2}{c}^2$.

Figure 4

(a) The final number of created positrons after the interaction with the laser field as a function of the laser’s photon energy $\hslash \omega$. The arrow indicates the minimal energy according to the threshold condition based on the bare energies. The vertical line is the predicted threshold based on the fully dressed bound state and continuum state energies. The leftmost curve corresponds to a nuclear potential with the bare ground state energy $E_{\alpha }=-0.6m{c}^2$, obtained for $V_0=1.85m{c}^2$, $D=3.885\lambda$, and $w=0.3\lambda$. The other two curves are for potentials $V(x)$ with higher bare ground state energies $E_{\alpha }=-0.5m{c}^2$, obtained for $V_0=1.8m{c}^2$, $D=3.428\lambda$, and $w=0.3\lambda$, and with $E_{\alpha }=-0.4m{c}^2$, obtained for $V_0=1.726m{c}^2$, $D=3.2\lambda$ and $w=0.3\lambda$. The laser amplitude was $\xi =0.5$ and the total interaction time $T=900\lambda /c$. (b) The dressed energy of the three ground states $E_{\alpha }(A_0,\omega )$ as a function of the photon energy $\hslash \omega$. The straight line is the prediction of the threshold energy according to Eq. (4.1), i.e., $m^{*}(\xi )c^2-2\hslash \omega$, such that the crossing with the curve $E_{\alpha }(A_0,\omega )$ corresponds to the minimal photon energy for the two-photon-based pair-creation process to occur.

Figure 5

Final energy spectra ${\rho }_{e+}(E)$ of the created positrons after the interaction of the vacuum state with a laser field of frequency $\omega =0.45m{c}^2/\hslash$ and scaled field strength $\xi =0.5$ (a) and $\xi =1$ (b). As a reference, the dashed vertical lines are the predicted peak positions based on the assumption that the created electron is trapped in a laser-dressed bound state. The interaction time was 50 laser cycles.

Figure 6

Final energy spectra of the created positrons after the interaction of the vacuum state with a laser field of frequency $\omega =0.55m{c}^2/\hslash$ and scaled field strength $\xi =0.5$ (a) and $\xi =1$ (b). As a reference, the dashed vertical lines are the predicted peak positions based on the assumption that the created electron is trapped in a laser-dressed bound state.

Figure 7

Final energy spectra of the created positrons after an interaction of the vacuum state with a laser field of scaled field strength $\xi =1$. As a reference, the dashed vertical lines are the predicted peak positions based on the assumption that the created electron is trapped in a laser-dressed bound state. Here the binding potential ($V_0=1.1m{c}^2$, $w=0.3\lambda$, and $D=3.2\lambda$) has only two bound states at energies $E_{\alpha }=0.1648m{c}^2$ and $E_{\beta }=0.6682m{c}^2$, which are in close resonance for the chosen photon energy $\hslash \omega =0.5m{c}^2$.