# Attosecond Streaking Time Delays: Finite-Range Interpretation and Applications
## 2 Finite-Range Interpretation
### Accumulation over Finite Time and Finite Range  
The formulas, Equations (7) and (8), give an interesting interpretation of the streaking time delay, namely that it can be viewed as the field-free time delay \(\Delta t_{\text{ff}}\), obtained by a photoelectron without the streaking field, weighted by the (instantaneous) streaking field strength as the photoelectron propagates through the potential. An important consequence of this interpretation is that the time delay is determined by the electron over a _finite time interval_ between the time of transition in the continuum \(t=t_{i}\) and the end of the streaking pulse at \(t=T_{s}\). During this time interval, the photoelectron explores a _finite range in space_ only.  
This interpretation has been tested in quantum streaking simulations for photoemission of an electron initially bound in a 1D model potential [20]. To illustrate the conclusions from that work, we consider here calculations using the model potential:  
\[V_{x}=-\frac{Z}{\sqrt{x^{2}+\alpha}}+V_{0}\exp^{-(|x|-x_{0})/\sigma)^{2}}, \tag{10}\]  
which consists of a (soft-core) Coulomb potential with \(\alpha=2.0\) centered at the origin and an effective charge \(Z=3\) and an additional Gaussian-shaped potential with depth \(V_{0}=-0.5\) a.u. and width \(\sigma=2.0\) centered at a distance \(x_{0}=850\) a.u. from the center of the Coulomb potential. Choosing the initial state to be the ground state of the Coulomb potential solutions of the corresponding 1D TDSE in the fields of a XUV pulse and an IR streaking pulse have been obtained and the streaking time delay has been extracted from the results (for methods to obtain streaking time delays from simulations, see e.g., [21]). Each of the calculations has been performed over a long propagation times such that the  
Figure 2: Streaking time delays for hydrogen atom (**a**); helium atom (**b**); and neon atom (**c**): a comparison of results of ab initio calculations is shown (open squares, taken from [24] for hydrogen and helium, [25] for neon), classical calculations (solid lines, Equation (8)) and predictions of analytical model formula (dashed lines, Equation (11)). For neon atoms, only the d-channel is considered in the analytical calculation.  
outgoing photoelectron wavepacket reaches distances of more than 1000 a.u., i.e., distances beyond the range of the Gaussian potential, at the end of the simulation.  
In Figure 3b, we compare the results for the streaking time delay with (stars) and without (circles) the additional Gaussian potential as a function of the number of cycles of the streaking pulse. For short streaking pulses, the delay is found to be essentially identical with and without the Gaussian potential. In each of these simulations with short pulses, the center of the outgoing wavepacket is located at the \(x<x_{0}=850\) a.u. at the end of streaking pulse (Figure 3a). In contrast, as soon as the center of the photoelectron wavepacket reaches the Gaussian potential, centered at \(x=850\) a.u., before the end of the pulse, the results from the simulations with and without Gaussian potential differ. Thus, the streaking time delay accounts for the presence of the Gaussian potential only when the electron wave packet reaches the potential before the interaction with the streaking pulse ceases, in agreement with the finite-range interpretation put forward based on the classical analysis.