Finite-range time delays in numerical attosecond-streaking experiments

We present results of numerical simulations and theoretical classical analysis of time delays with respect to the instant of ionization in a numerical streaking experiment. We show that the time delay is related to a finite range in space, which the emitted electron probes after its transition into the continuum until the streaking pulse ceases. This finite-range time delay results from the coupling of the atomic potential and the streaking field and strongly depends on the parameters, in particular the duration, of the streaking field. It can be represented as an integral or sum over piecewise field-free time delays weighted by the ratio of the instantaneous streaking field strength relative to the field strength at the instant of ionization.

Figure 1

(a) $\Delta t_s$ as a function of the position $x_0$ of the Gaussian potential. We compare results of TDSE (stars, squares, and circles for $V_0=-0.5,-2.0$, and $-4.0$, respectively) with those of classical calculations, Eq. (7) (lines). Also shown is the TDSE result without Gaussian potential (diamond on the right end). In the classical calculations the parameter $\alpha$ varied between $0.985$ and $1.084$. (b) Comparison of $\Delta t_s$ from TDSE for different cycle numbers of the streaking pulse with (solid line with circles) and without (dashed line with squares) the Gaussian potential. The Gaussian potential was located at $x_0=650$. Other parameters are given in the text.

Figure 2

Classical predictions for the time delay as a function of the delay between XUV ionizing and IR streaking pulses for different fitting parameter $\alpha$ by using Eq. (7). Calculations are performed for the 1D potential $V_{\mathrm{CG}}\left(x\right)$ in Eq. (1) with $Z=3.0$, $a=2.0$, $V_0=-0.5$, $\sigma =2.0$, and $x_0=20$.

Figure 3

Classical estimates for the “field-free” delay distance of an electron, released at the peak of a three-cycle streaking pulse (wavelength of 800 nm) in the 1D Coulomb potential [i.e., $V_0=0$ in Eq. (1)], as a function of the XUV photon energy. This distance was determined by the position of the electron at the time instant when the streaking field changed to $99\%$ (solid line) and $95\%$ (dashed line) of the peak field strength.

Figure 4

Results of numerical simulations for the streaking time delays $\Delta t_s$ (solid line with circles and diamonds) as a function of (a) the range $x_p$ of a short-range potential, (b) the frequency of the ionizing XUV field, and (c) the wavelength of the streaking pulse are compared with those for the WS time delay [dashed lines with squares in (a) and (b) and solid circles in (c)]. Laser parameters are: $I_{\mathrm{XUV}}=1\times {10}^{15}$ W/cm${}^2$, $T_{\mathrm{XUV}}=600$ as, ${\omega }_{\mathrm{XUV}}=100$ eV [(a),(c)], ${\varphi }_{\mathrm{XUV}}=-\pi /2$, $I_s=1\times {10}^{12}$ W/cm${}^2$, $N_s=3$ cycle [(a),(b)], $T_s=32.02$ fs [(c)], ${\lambda }_s=800$ nm [(a),(b)], and ${\varphi }_s=-\pi /2$.

Figure 5

Comparison of time delays from 3D quantum streaking simulations (open circles, extracted from Ref. [13]) with the results of the present classical approximations, Eq. (7) (solid lines) and Eq. (8) (dashed lines). For both potentials the initial states are chosen to be the $1s$ state. The parameter $\alpha$ varied between $1.003$ and $1.067$.