Interferences of real trajectories and the emergence of quantum features in electron-atom scattering in a strong laser field

Using the example of electron-atom scattering in a strong laser field, it is shown that the oscillatory structure of the scattered electron spectrum can be explained as a consequence of the interference of the real electron trajectories in terms of Feynman’s path integral. While in previous work on quantum-orbit theory the complex solutions of the saddle-point equations were considered, we show here that for the electron-atom scattering with much simpler real solutions a satisfactory agreement with the strong-field-approximation results can be achieved. Real solutions are applicable both for the direct (low-energy) and the rescattering (high-energy) plateau in the scattered electron spectrum. In between the plateaus and beyond the rescattering cutoff good results can be obtained using the complex (quantum) solutions and the uniform approximation. The interference of real solutions is related to the recent attosecond double-slit experiment in time.

Figure 1

(Color online) The differential cross section for potential scattering of electrons having the initial kinetic energy $E_{{\mathbf{k}}_i}=18\mathrm{eV}$ on He atoms in the presence of a monochromatic linearly polarized laser field, as a function of the number of photons $n$ exchanged with the laser field. The laser wavelength is $1060\mathrm{nm}$ and the intensity $2.6\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. The incident electron momentum is in the direction of the laser field polarization, and the forward scattering is considered. The direct (D) and the rescattering (R) plateaus with the corresponding cutoffs at $n_{\mathrm{D}}=79$ and $n_{\mathrm{R}}=214$ are denoted in the figure. The results obtained using the strong-field approximation (SFA) are compared with the results obtained using the uniform approximation (UA) with the real solutions of the stationary-point equations for the direct and the rescattering plateau and with the complex solutions in the transition region between the plateaus (C).

Figure 2

(Color online) A comparison of the numerical results for the direct part of the differential cross section obtained using different methods. The laser, atomic, and the incident electron parameters are the same as in Fig. 1.

Figure 3

(Color online) Trajectories of the electron in the laser field for the parameters of Fig. 1. The direct-scattering trajectories (solid and dashed line) for $n=60$ and the rescattering trajectories (dotted and dot-dashed line) for $n=200$ are presented. The rescattering trajectories correspond to the solutions having the highest maximum in Fig. 4, denoted by $p=1$.

Figure 4

(Color online) The solutions of the stationary-point equations (18, 19) for the parameters of Fig. 1. The number of photons $n$ exchanged with the laser field is presented as a function of the travel time $\tau$, expressed in optical periods $T$. The solutions for the direct scattering (dashed line, denoted by D) are presented as a function of the first scattering time $t$, $0<t∕T<1$. For the rescattering there are more solutions that are classified by the numbers $j$ and $p$, as explained in the text.

Figure 5

(Color online) The rescattering part of the differential cross section as a function of the number of photons $n$ exchanged with the laser field for the same laser, atomic, and the incident electron parameters as in Fig. 1. A comparison of the strong-field approximation (SFA—solid line) and the stationary phase approximation (SPR—dashed line).