Influence on the ionization yields

Representing a recent accessible quantity and providing information about contributing processes the final ionization yields are discussed as a function of intensity for finite pulses with a negative detuning. In section III.3 a transition between quadratic and linear intensity scaling was found for a resonant excitation due to the onset of Rabi

2As an example in figure IV.3 (d) forhδ ω¯ =−1.63eV atI=1.72·10¹⁴W/cm²the ionization dynamics of the complete TDSE and the reduced atom with one excited single particle state show almost no quantitative differences.

oscillations. It is close at hand to verify this feature for an off-resonantly driven atom due to the following additional aspects. For a resonant excitation the quadratic scaling at low intensities of the ionization yields could be almost exclusively attributed to a sequential two-photon ionization in which the atom absorbs one photon to transfer an electron to the first excited state and afterwards a second photon for the transition into the continuum. In contrast, one has to recall that in the off-resonant case the direct two-photon process also contributes to the ion yields and is characterized by a quadratic intensity scaling. These additional ionization paths are enhanced beside the Rabi processes because the Rabi oscillations do not completely deplete the ground state.

Therefore, it is not ensured that the presence of Rabi oscillations leads automatically to a linear intensity scaling as in the case of a resonant laser excitation.

In figure IV.6 each plot shows the ionization yields in dependence of the intensity for different pulse durations at a fixed detuning. The envelope of the pulse is a rectangle with smoothed edges during the first and the last five field oscillations. Decrease and rise follow the right and left edge of a Gaussian curve withσ²=5T_L/3.5. T_L represents the period of one field oscillation. For a negative detuning ofhδ ω¯ =−0.27eV, displayed in figure IV.6 (a), the curves for all pulse durations clearly exhibit a crossover between different intensity scalings. Also the position of the transition moves to a lower intensity for increasing pulse duration forhδ ω¯ =−0.27eV and pulse durations of 3 fs (blue line), 6 fs (red line) and 12 fs (green line). Similar to a resonant excitation, this behaviour reflects the onset of Rabi oscillations at a pulse area ofA=π. However, in the curve for

¯

hδ ω=−0.27eV and a pulse duration 36 fs (orange line) the intensity of the crossover does not shift to smaller intensities as expected. Compared to the ion yields of the 12 fs pulse the transition takes place at roughly the same intensity. A similar behaviour is found for a detuning ofhδ ω¯ =−0.82eV (fig. IV.6 (b)). At least for pulse durations of 3 fs, 6 fs and 12 fs one observes an intensity scaling below a quadratic scaling before saturation effects become of importance. Nevertheless a strict linear regime is not established. Only by rising the pulse duration from 3 fs to 6 fs the weak crossover shifts as expected for the onset the Rabi cycles. Forhδ ω¯ =−1.63eV (fig. IV.6 (c)) the ion yields do not reveal a transition between different intensity scalings. One can summarize that for an increasing absolute value of the detuning and the pulse duration the crossover loses its sharpness and is not correlated with the pulse area.

This observation reflects the competition between the two ionization pathways con-tributing to the off-resonantly driven atom, on the one hand the direct two-photon ionization and on the other hand the Rabi-assisted ionization. Which process dominates critically depends on the characteristics of the Rabi cycles in the off-resonant case. These are the dependence of the amplitude of the oscillations on the electric field strength and the contribution of the detuning to the Rabi frequency. In particular, the latter leads to a limiting value of the Rabi frequency for low intensities. This value defines a critical pulse duration at which for all intensities at least one Rabi cycle should be present, even though with a very small amplitude.

Elim0→0Ω=δ ω ⇒t^c_p= 2π

δ ω. (IV.3)

10⁻⁴

Figure IV.6.:Ionization yields after a laser pulse as a function of intensity for the detuningshδ ω¯ =

−0.27eV (a),hδ ω¯ =−0.82eV (b) andhδ ω¯ =−1.63eV (c) obtained via the numerical solution of the complete time-dependent Schrödinger equation. The corresponding pulse durations are 36 fs (orange dashed line), 12 fs (green dotted line), 6 fs (red dashed dotted line) and 3 fs (blue solid line). The grey lines mark a strict linear or quadratic intensity scaling andt^c_pdenotes the maximal Rabi period in the case ofE0=0for each detuning, respectively.

The critical pulse lengths for the applied detunings are given in the corresponding plots of figure IV.6. Obviously, for pulse duration equal or larger than the critical value, the position of the transition between the linear and quadratic intensity scaling is no longer exclusively determined by the pulse duration and the corresponding pulse area. Instead, only the intensity has to be large enough to realise a notable amplitude of the Rabi oscillations resulting in a linear intensity scaling. This behaviour is demonstrated in figure IV.6 (a). Ion yields are plotted for pulse durations which are almost equal (green line, 12 fs) as well as larger (orange line, 36 fs) than the critical value (t_p^c=15fs). In agreement with the previous considerations the position of the crossover between the quadratic and the linear intensity scaling is found for both pulse durations approximately at the same intensity. One recognizes a smooth crossover for pulse lengths above the critical value, because the sufficient amplitude of the Rabi oscillations to establish a linear intensity scaling cannot be declared in a precise condition. In contrast, for the results, at which the critical pulse area argument holds, the transition is rather sharp.

Therefore the slope of the curve first starts to deviate slightly from the quadratic scaling and bends itself over a large intensity interval until the Rabi oscillations finally dominates and the linear scaling regime occurs.

For hδ ω¯ =−0.82 eV and hδ ω¯ =−1.63 eV in figures IV.6 (b) and (c) the ion yields deviate from the quadratic intensity scaling, but do not access the regime of a purely linear intensity scaling. A larger detuning results in a smaller critical pulse duration enabling Rabi oscillations for almost all applied pulses. Nevertheless, also a higher field strength is required to excite Rabi cycles which induce deviations from the quadratic intensity scaling. Thus, the direct two-photon ionization dominates until the intensity is sufficient to provide an adequate amplitude of the Rabi oscillations between the ground state and the first excited state. The results manifest that forhδ ω¯ =−0.82eV

andhδ ω¯ =−1.63 eV the signal already saturates before the Rabi cycles dominate the ionization. Therefore the ionization scales with an exponent smaller than two without reaching a strict linear intensity dependence. The characteristic intensity, at which deviations from the quadratic intensity scaling for pulse lengths larger than the critical pulse length occur, is related to the amplitude of the Rabi oscillations and depends on the detuning. Figure IV.6 indicates that for an increasing detuning the position of the crossover shifts to higher intensities, as expected.