Attosecond-streaking time delays: Finite-range property and comparison of classical and quantum approaches

We theoretically study time delays obtained using the attosecond-streaking technique. To this end, we compute time delays by numerically solving the corresponding time-dependent Schrödinger equation and analyze the delays using two classical methods, namely, a perturbative approach and a full numerical solution of Newton's equation describing the motion of the photoelectron in the continuum. A good agreement between the quantum streaking results and those from the full classical solution is found. This indicates that the streaking time delay arises from the continuum dynamics of the electron in the coupled potential of the Coulomb and streaking fields, while the transition of the photoelectron from the bound state to the continuum occurs instantaneously upon absorption of the photon. We further analyze the variation of the time delay with respect to the delay between the ionizing XUV pulse and a long streaking pulse, its dependence on the polarization direction of the streaking pulse, and the influence of the shape of the streaking pulse and/or additional static electric fields on the numerically obtained time delays. The results are interpreted based on the previously revealed property that the attosecond-streaking time delay depends on the finite region in space over which the electron propagates between its instant of transition into the continuum and the end of the streaking pulse.

Figure 1

(a) Streaking traces obtained from numerical simulations for photoemission in the 1D Coulomb potential $V_{\mathrm{C}}\left(z\right)$. An eight-cycle streaking pulse with $I_s=1\times {10}^{12}$ W/cm${}^2$, ${\lambda }_s=800$ nm, and ${\varphi }_s=-\pi /2$ was used in the calculations. We compare the streaking trace obtained from the TDSE [solid (blue) line] with that from the original streaking formula [Eq. (4); dashed (green) line]. (b) An enlargement of (a) to show the momentum shift $\Delta k$, the streaking time delay $\Delta t_s$, and the relation between them.

Figure 2

Comparison of streaking time delays from quantum streaking simulations (black circles) with classical results from the perturbative approach [dashed (blue) lines] and the full numerical solution [solid (red) lines]. In our analysis we considered three potentials: (a) the 1D Coulomb potential [$V_{\mathrm{C}}\left(z\right)$], (b) the combination of 1D Coulomb and Gaussian potentials [$V_{\mathrm{CG}}\left(z\right)$], and (c) the 3D Coulomb potential [$V\left(\mathit{r}\right)$].

Figure 3

Relative difference in the momentum shift $\Delta k$ calculated classically from the perturbative approach vs the full numerical solution for $V_{\mathrm{C}}\left(z\right)$ [solid (blue) line with circles] and $V_{\mathrm{CG}}\left(z\right)$ [$z_0=200$; solid (green) line with asterisks]. The relative difference is defined as $|(\Delta k^{\left(\mathrm{pert}\right)}-\Delta k^{\left(\mathrm{numerical}\right)})/\Delta k^{\left(\mathrm{numerical}\right)}|$. We have considered photoemission of an electron with a final asymptotic momentum of 2.0, which is streaked by a three-cycle, 800-nm laser pulse.

Figure 4

Two-dimensional model potentials, as defined (a) in Eq. (19) and (b) in Eq. (20), plotted on a logarithmic scale as $log[-V(x,y\left)\right]$.

Figure 5

Time delay as a function of streaking-pulse cycle number for the 2D potentials defined (a) in Eq. (19) and (b, c) in Eq. (20) and polarizations of the streaking (and coaligned XUV ionizing) field in the $x$ direction [(green) line with asterisks], at ${45}^{\circ }$ [(cyan) line with crosses], and in the $y$ direction [(red) line with squares]. The results are compared with those for the pure 2D Coulomb potential without the additional potential [(blue) line with circles] and a streaking field polarized in the $x$ direction. In (c), the original streaking time delays for the 2D potential, Eq. (20) as shown in (b), have been shifted to match the result for the shortest streaking pulse for the pure 2D Coulomb potential.

Figure 6

Streaking field with a pedestal [solid (red) line]: $E_0=5.34\times {10}^{-3}$ (i.e., $I_s=1\times {10}^{12}$ W/cm${}^2$), $T_s=331$, ${\beta }_p=0.2$, and $T_p=750$. For comparison, a basic three-cycle, 800-nm streaking pulse [dashed (blue) line] is also shown.

Figure 7

Streaking time delay as a function of (a, b) the pedestal length $T_p$ (${\beta }_p=0.2$) and (c, d) the pedestal strength ${\beta }_p$ ($T_p=750$). (a, c) Results for $Z=1.0$; (b, d) results for $Z=3.0$. The influence of an additional static field on the streaking time delay is also present [dashed (green) lines with asterisks and crosses] in (a) and (b).

Figure 8

Streaking time delays as a function of the strength of the additional static field for a basic three-cycle, 800-nm streaking field for two potentials: $Z=1.0$ [solid (blue) line with circles] and $Z=3.0$ [solid (green) line with squares].