# Attosecond Streaking Time Delays: Finite-Range Interpretation and Applications
## 2 Finite-Range Interpretation
### Analytical Formula  
Based on the interpretation that the streaking time delay is accumulated by the photoelectron over a finite time only, and hence can be understood as a probe of the (long-range) atomic potential over a finite distance, one can proceed to consider cut-off potentials for a further analysis of the time delay. In order to do so, one may assume a classical-quantum correspondence for the field-free time delay, in which the classical estimate, Equation (8), is decomposed in a sum as [23]:  
\[\Delta t_{s}\simeq\sum_{j=1}^{N}\frac{E_{s}(t_{j})}{E_{s}(t_{i})}\left[\Delta t _{short}^{(j)}+\Delta t_{phase}^{(j)}+\Delta t_{log}^{(j)}\right]\, \tag{11}\]  
where the terms in the brackets are the well-known short-range, Coulomb-phase and logarithmic contributions to the quantum field-free time delay. Note that each of the terms represents the derivative of a contribution to the quantum mechanical phase shift with respect to the photoelectron's energy.  
For example,  
\[\Delta t_{phase}(k,r)=\frac{1}{k}\frac{\partial}{\partial k}\text{arg}\left[ \Gamma(1+l+i\eta)\right] \tag{12}\]  
and  
\[\Delta t_{log}(k,r)=\frac{1}{k}\frac{\partial}{\partial k}\left[\frac{Z}{k} \text{ln}(2 kr)\right]\, \tag{13}\]  
where \(l\) is the angular momentum of the photoelectron, \(\eta=-Z/k\), and \(Z\) is the charge of the residual ion. Specifically, \(\Delta t_{short}\) is the contribution related to the short-range part of the atomic potential,  
Figure 3: (**a**) Range of wavepacket propagation in an 800 nm streaking pulse after emission at the peak of the pulse as a function of the number of cycles for 1D soft-core model; (**b**) streaking time delay in a potential with additional perturbation (stars) at \(x_{0}=850\) a.u. as compared to the delay obtained without the additional potential (circles) as a function of the number of cycles.  
while the other two terms result from the scattering off the \(-Z/r\) part of the ionic potential for nuclear charge \(Z\). Note that, for an infinite range, Coulomb potential \(\Delta t_{log}\) diverges, while \(\Delta t_{phase}\) converges. Therefore, one typically refers to the sum of the short-range and Coulomb-phase term as the widely known Wigner-Smith time delay \(\Delta t_{WS}\) in the context of scattering off an ionic core with a long-range Coulomb potential [6; 7].  
Each of the three contributions in Equaiton (11) can then be again written as a difference  
\[\Delta t_{i}^{(j)}=\Delta t_{i}(k_{j+1},r_{j+1})-\Delta t_{i}(k_{j},r_{j})\, \tag{14}\]  
where \(\Delta t_{i}(k,r)\)\((i=\text{short},\text{phase},\text{log})\) is the respective contribution to the field-free time delay accumulated during the propagation from the origin of the transition into the continuum to location at a _finite_ distance \(r\) with momentum \(k=k(r)=\sqrt{2(E_{asym}-V(r))}\). Thus, the terms \(\Delta t_{i}(k,r)\) correspond to time delay contributions in a potential which is cut-off at \(r\). Consequently, due to the finite-range characteristic, none of the terms, including the logarithmic term, diverge here. This has the advantage that the corresponding phase-shift in the wavefunction is finite and one does not encounter conceptual issues like the divergence of the logarithmic term in a Coulomb potential.  
It has been shown [23] that each of the terms in Equation (11) can be evaluated analytically for an atom using standard methods, such as the distorted Born approximation and the single-active-electron approximation. Estimates based on the analytical formula (dashed lines, Equation (11)) are found to be in good agreement with the results of ab initio simulations and those of classical calculations for the streaking time delay (see Figure 2, [23]). This validation puts us in the position to use this analytical formula for further analysis of certain aspects of the streaking time delay.