Controlling electron motion with attosecond precision by a shaped femtosecond intense laser pulse

We propose a temporal double-slit interferometric scheme to characterize the shaped intense femtosecond laser pulse directly, which can be applied to control electron tunneling wave packets with attosecond precision. By manipulating the spectral phase of the input femtosecond pulse in the frequency domain, one single pulse is split into two subpulses and their waveforms can be controlled by adjusting the spectral phase. In the interaction between the shaped pulse and atoms, the two subpulses are analogous to Young's double slit in the time domain. The interference pattern in the photoelectron momentum distribution can be used to retrieve the peak electric field and the time delay between two subpulses. Based on the characterization of the shaped pulse, we demonstrate that the subcycle dynamics of photoelectrons can be controlled with attosecond precision. The feasibility of this scheme is confirmed through quantum-trajectory Monte Carlo simulations and numerical solutions of the three-dimensional time-dependent Schrödinger equation.

Figure 1

Sketch of the maps for ionization processes if Xe atoms are exposed to (a) the unshaped single pulse and (b) the shaped pulse with a time delay of $\mathrm{\Delta }t$ between the two subpulses. The PMDs are simulated by the QTMC method. The laser intensity of the input unshaped pulse is $I_0=1.5\times {10}^{14}$ W/${\mathrm{cm}}^2$, the central wavelength is ${\lambda }_0=800$ nm, and the pulse duration is 20 fs.

Figure 2

(ai) Electric field $F\left(t\right)$ of the shaped pulse as a function of time. (aii) Electric field and negative vector potential $-A\left(t\right)$ within one optical cycle indicated by the violet dashed rectangle shown in (ai). Here $t_{i1}$ and $t_{i2}$ denote the ionization times of intracycle interference trajectories which have the same final momentum with a time difference $\delta t$. Simulated PMDs by QTMC are obtained by considering electrons ionized at different time ranges: (b) $t_i<0$, (c) $t_i$ in a single optical cycle enclosed by the dashed rectangle, (d) $t_i$ in green regions, (e) $t_i$ in green and blue regions, and (f) $t_i$ in green, blue, and red regions. (g) The PMD calculated by the TDSE. The white dashed lines are used to indicate the subring and jetlike structure. Here ${\lambda }_s=780$ nm and the other pulse parameters are the same as in Fig. 1.

Figure 3

Energy spectra for shaped pulses simulated by the (a) QTMC and (b) TDSE methods and the corresponding (c) extracted time delay $\mathrm{\Delta }t$ and (d) peak-electric-field strength $F_0$ of the shaped pulse at different ${\lambda }_s$. The actual values of $\mathrm{\Delta }t$ and $F_0$ are also presented for comparison. The arrows in (a) and (b) are used to indicate the subring used in Fig. 4.

Figure 4

Angular distribution corresponding to a specific subring at different ${\lambda }_s$ simulated by the (a) QTMC and (b) TDSE methods. The positions for peaks p1 and p2 are indicated by vertical lines. (c) Curves obtained from Eq. (18), illustrating how the ionization time difference $\delta t$ between the two interfering electron trajectories contributing to the two peaks (labeled p1 and p2) in (a) and (b) can be extracted from the curves. (d) Extracted ${\lambda }_s$-dependent $\delta t$ corresponding to peaks p1 and p2. The actual values of $\delta t$ obtained through the statistics of the trajectories in the QTMC simulations are also presented for comparison. The subrings used above are indicated by arrows in Figs. 3 and 3.

Figure 5

(a) PMD calculated by the QTMC method including the focal-volume-averaging effect (FAE), where ${\lambda }_s=780$ nm and the peak laser intensity is $1.5\times {10}^{14}$ W/${\mathrm{cm}}^2$. The corresponding ${\lambda }_s$ dependences of (b) the extracted time delay $\mathrm{\Delta }t$, (c) the peak-electric-field strength $F_0$ of the shaped pulse, and (d) $\delta t$ corresponding to peaks p1 and p2 are compared with the actual values.