## 2 Results  
For a set of delays \(\{d_{i}\}\), \(i=0..n\), let us consider two delay scans \(\mathbf{R}=\{R_{0},\ldots,R_{n}\}\) and \(\mathbf{S}=\{S_{0},\ldots,S_{n}\}\) each consisting of a sequence of Ne\({}_{2}^{+}\) ion-TOF signals, reduced as explained in Section 4, and padded with zeros outside of the delay range scanned. We will consider \(\mathbf{R}\) our reference scan, which for convenience is assumed to be noise-free. The definition of cross-correlation without normalization, \(\Gamma_{\mathbf{RS},q}\) is:  
\[\Gamma_{\mathbf{RS},q}=\sum_{i}R_{i}S_{i+q}\equiv(\mathbf{R}\star\mathbf{S})_ {q} \tag{1}\]  
Note that \(\Gamma\) is linear in the two sequences (scaling either one by a factor \(\alpha\) scales \(\Gamma\) by the same factor), and that \(\Gamma_{\mathbf{S},q}\) is the autocorrelation of \(\mathbf{S}\), which has a maximum for \(q=0\):  
\[(\mathbf{S}\star\mathbf{S})_{0}=\sum_{i}(S_{i})^{2} \tag{2}\]  
We now assume that any scan \(\mathbf{N}=\{N_{0},\ldots,N_{n}\}\) consisting of pure noise has zero (negligible) cross-correlation with any other scan (including itself except, obviously, at zero-shift), thus:  
\[(\mathbf{N}\star\mathbf{N})_{q} = \delta_{q0}\sum_{i}(N_{i})^{2} \tag{3}\] \[(\mathbf{R}\star\mathbf{N})_{q} = 0 \tag{4}\]  
with \(\delta_{q0}\) the Kronecker delta; we use Equation (3) to define a variance \(\sigma^{2}=(\mathbf{N}\star\mathbf{N})_{0}=\sum_{i}(N_{i})^{2}\). Equations (3) and (4) strictly hold for white noise and infinite \(n\); in a real-life situation we can expect that noise correlation just decays much faster than signal correlation: thus all scans whose autocorrelation is sharply peaked are probably pure noise, and will also have poor cross-correlation with the reference scan. Any noisy scan \(\mathbf{S}\) can be written in terms of the reference scan \(\mathbf{R}\) and a pure-noise sequence \(\mathbf{N}\) as: \(\mathbf{S}=\alpha\mathbf{R}+\mathbf{N}\), with \(\alpha\) a real number. In a worse-case scenario, \(S_{i}\) may be shifted (for simplicity by an integer index \(r\)), that is: \(S_{i}=\alpha R_{i+r}+N_{i}\). Then:  
\[(\mathbf{R}\star\mathbf{S})_{q}=\alpha(\mathbf{R}\star\mathbf{R})_{q-r}+( \mathbf{R}\star\mathbf{N})_{q}=\alpha(\mathbf{R}\star\mathbf{R})_{q-r} \tag{5}\]  
note that \(\mathbf{R}\star\mathbf{S}\) is proportional to \(\mathbf{R}\star\mathbf{R}\) but shifted by \(r\) as expected; this is the argument invoked in Ref. [5] to associate the position of the maximum of the cross-correlation function to the shift of the peak being sought. In our work [1] it was not necessary to include a shift; while Equation (5) could be used to estimate the signal amplitude \(\alpha\), it is preferable to use the autocorrelation instead:  
\[(\mathbf{S}\star\mathbf{S})_{q}=\alpha^{2}(\mathbf{R}\star\mathbf{R})_{q}+( \mathbf{N}\star\mathbf{N})_{q}=\alpha^{2}(\mathbf{R}\star\mathbf{R})_{q}+ \sigma^{2}\delta_{q0} \tag{6}\]  
(note that the shift by \(r\) cancels, here). Equation (6) tells us in particular that all autocorrelation sequences should be the same except for a scale factor \(\alpha^{2}\) and a sharp noise peak at \(q=0\). Visual inspection shows that this is true for a number of scans which we consider good; other scans exhibit a narrower, or structured, sometimes negative, autocorrelation, probably indicating correlated noise, i.e., a drift during the measurement; the rest only exhibits the noise peak, indicating no signal at all (Figure 1). From now on we will consider \((\mathbf{R}\star\mathbf{R})\) to be noiseless and, as empirically found, satisfactorily approximated by a gaussian; we derive the width of the gaussian from our best scan (Figure 1). Our choice of a gaussian is purely empirical, and our method does not critically depend on it: to the extent that all good scans have the same shape of the autocorrelation function, Equation (6), the best fit parameters will be the same for all good scans, except for a scale factor for the height.