## Theory
### Electron Streaking  
In electron streaking, an ionization pulse, typically in the VUV to X-ray region, ionizes atoms in the presence of an AC streak field [13], and the energy spectrum of the photoelectrons released in the process is measured. With no streak field present and a sufficiently flat ionization cross section,the spectrum of the X-ray pulse is mapped to the photoelectron spectrum. When the streak field is on, the photoelectrons gain or lose kinetic energy depending on their time of birth relative to the streak field oscillations. Recording photoelectron spectra as a function of time delay between the X-ray pulse and the streak field results in a two-dimensional time-frequency distribution, which exhibits wiggles that are synchronized to the streak field oscillations. Ideally, the two-dimensional distribution is formally identical to the spectrogram (1). In order to compute the photoelectron spectrum, first, we calculate the transition amplitude, \(a_{\vec{p}}\), from the ground state to a continuum state characterized by the momentum, \(\vec{p}\). For the sake of simplicity, we use the single active electron approximation, e.g., as published in reference [14] and references therein. At times large enough for the ionization field to vanish, the resulting transition amplitude obtained by means of a first-order perturbation theory is  
\[a_{\vec{p}}=-\frac{\mathrm{i}}{\hbar}\int\limits_{-\infty}^{\infty}\mathrm{d}t \,\vec{d}\left[\vec{p}\,\right]\vec{E}_{X}(t)\,\exp\left[\frac{\mathrm{i}}{ \hbar}\left(\frac{\left|\vec{p}\,\right|^{2}}{2m}+I_{p}\right)t\right]. \tag{2}\]  
Here, \(\vec{E}_{X}(t)\) is the X-ray electric field, \(\vec{d}\left[\vec{p}\,\right]\) is the dipole transition matrix element from the ground to the continuum state, \(\left|\vec{p}\,\right|^{2}/(2m)\) is the energy of the final continuum state and \(I_{p}\) denotes the ionization potential of the atom. The corresponding photoelectron spectrum is  
\[S(\left|\vec{p}\,\right|)\propto\left|\int\limits_{-\infty}^{\infty}\mathrm{d}t \,\vec{d}\left[\vec{p}\,\right]\vec{E}_{X}(t)\,\exp\left[\frac{\mathrm{i}}{ \hbar}\left(\frac{\left|\vec{p}\,\right|^{2}}{2m}+I_{p}\right)t\right]\right|^ {2}. \tag{3}\]  
From Equations (2) and (3), it can be seen that the spectrum resulting from the transition amplitude, \(a_{\vec{p}}\), is directly connected to the X-ray spectrum, both in phase and amplitude. When the streak field temporally overlaps with the emerging electron wave packet, the photoelectrons are either accelerated or decelerated, and the time dependent vector potential is imprinted on the photoelectron spectrum. This scenario is treated as a strong field approximation where the ionic potential is neglected with respect to the streak field. The transition amplitude, \(a_{\vec{p}}\left(\tau\right)\), now a function of time delay, \(\tau\), between the two fields, is given by  
\[a_{\vec{p}}\left(\tau\right) = -\frac{\mathrm{i}}{\hbar}\int\limits_{-\infty}^{\infty}\mathrm{d}t \,\vec{d}\left[\vec{p}+e\vec{A}_{S}(t)\right]\vec{E}_{X}(t-\tau) \tag{4}\] \[\exp\left\{\frac{\mathrm{i}}{\hbar}\left[I_{p}t-\frac{1}{2m}\int \limits_{t}^{\infty}\mathrm{d}t^{\prime}(\vec{p}+e\vec{A}_{S}(t^{\prime}))^{2} \right]\right\}.\]  
where \(e\) denotes the electron charge, \(\vec{p}+e\vec{A}_{S}(t)\) is the instantaneous momentum of a free electron in the streak field and \(\vec{A}_{S}(t)\) is the vector potential associated with the streak field in the Coulomb gauge with the electric field vector, \(\vec{E}_{S}(t)=-\partial\vec{A}_{S}/\partial t\). Without the streak field, i.e., \(\vec{A}_{S}(t)=0\), Equation (4) reduces to Equation (2).