Electron Dynamics Driven by Light-Pulse Derivatives

We demonstrate that ultrashort pulses carry the possibility for a new regime of light-matter interaction with nonadiabatic electron processes sensitive to the envelope derivative of the light pulse. A standard single pulse with its two peaks in the derivative separated by the width of the pulse acts in this regime like a traditional double pulse. The two ensuing nonadiabatic ionization bursts have slightly different ionization amplitudes. This difference is due to the redistribution of continuum electron energy during the bursts, negligible in standard photoionization. A time-dependent close-coupling approach based on cycle-averaged potentials in the Kramers-Henneberger reference frame permits a detailed understanding of light-pulse derivative-driven electron dynamics.

Figure 1

Nonadiabatic ionization. (a) A flattop laser pulse rising ($\uparrow$) and falling ($\downarrow$) over a time span of $T/2=25.5\mathrm{a}.\mathrm{u}.$ between the maximum of the electric field amplitude $F(t)$ and its maximal derivative. These two Gaussian half pulses, which act similarly as a pump and probe pulse in the nonadiabatic regime, encompass a plateau of duration $T_c$. (b) Electron spectra for two energies as a function of plateau length $T_c$ for an electron initially bound by a weak potential at energy $E_g=-0.0277\mathrm{a}.\mathrm{u}.$ exposed to a laser pulse of peak amplitude $F=0.5\mathrm{a}.\mathrm{u}.$ and frequency $\omega =0.314\mathrm{a}.\mathrm{u}.$; see the text. TDSE solutions with the full Hamiltonian are given as lines and with $H_0$ from Eq. (2) as symbols. (c) Nonadiabatic electron spectrum $P_E(T_c)$. The lines mark the maxima and minima predicted from Eq. (4). (d) Phase difference ${\phi }_E$, extracted by fitting Eq. (3) to spectra for fixed $E$ as in (b) (symbols); from benchmark calculations obtained through wave-packet partitioning (see the text) for the Gaussian pulse, i.e., $T_c=0$ (solid line); and approximated with Eq. (7) (dashed line).

Figure 2

Nonadiabatic electron spectra from Gaussian half-pulses. (a) Sketch of the partitioning approach for the rising ($\uparrow$) half-pulse. (b) The artificial laser field with the falling pulse length $T_{\mathrm{ad}}$ long enough to not affect nonadiabatic dynamics while the rising part has as before $T/2=25.5\mathrm{a}.\mathrm{u}.$ (Note that for the figure sketch $T_{\mathrm{ad}}=170\mathrm{a}.\mathrm{u}.$, while in the calculations $T_{\mathrm{ad}}=850\mathrm{a}.\mathrm{u}.$ has been used, which is sufficiently long to obtain converged results.) (e) The photoelectron spectrum from the partitioning approach as sketched in (a) (lines) and from the artificial laser field as in (b) (symbols). (c), (d), and (f) show information analogous to (a), (b), and (e), respectively, but for the falling ($\downarrow$) half-pulse instead of the rising half-pulse. The peak field strength and laser frequency are the same as in Fig. 1.

Figure 3

Energy reshuffling of continuum electron wave packets. Photoelectron spectra from the rising (red) and falling (blue) half-pulses. Results are obtained by the CC equations (6) with (solid and dashed lines) and without (dashed-dotted lines) the continuum coupling contribution $M_{E{E}^{\prime }}$, as well as with wave-packet partitioning (symbols). The laser parameters are the same as in Fig. 1.

Figure 4

The nonadiabatic electron spectrum of a single Gaussian pulse of width $T=51\mathrm{a}.\mathrm{u}.$ (dashed-dotted line) and combined from the sequence of a rising and falling half-pulse with $T/2=25.5\mathrm{a}.\mathrm{u}.$ (open circles). In addition the contribution from the first electron burst ${{\tilde{A}}_{\uparrow }}^2(E)$ (solid, red line) and the second one ${A_{\downarrow }}^2(E)$ (dashed blue line) are shown. The inset reveals Stueckelberg oscillations [21] of the normalized spectrum, ${\overline{P}}_E\equiv P_E/[2{{\tilde{A}}_{\uparrow }}^2(E)+2{A_{\downarrow }}^2(E)]$. The laser parameters are the same as in Fig. 1.

Figure 5

Photoelectron spectrum ${A_{\uparrow }}^2(E)$ of a rising half-pulse only as in Fig. 2 (solid red line) and ${{\tilde{A}}_{\uparrow }}^2(E)$ modified by the falling half-pulse for different plateau lengths $T_c$ ($T_c=0$, dashed black line; $T_c=2000\mathrm{a}.\mathrm{u}.$, dashed-dotted blue line). The laser parameters are the same as in Fig. 1.