# Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective
## 7 Applicability of the Frozen-Density Approximation  
Let us now consider the treatment of electron dynamics during the interaction of an electronic system with the probe pulse. In all examples discussed in the previous Sections, duration of a probe pulse was chosen such short that changes in the electron density during the probe were negligible. As a result, the frozen-density approximation, which neglects electron dynamics during the interaction with the probe pulse, could be applied for the QED derivation of Equation (18) for time-resolved NRXS, of Equation (30) for time-resolved RXS, and of Equation (42) for the time-resolved photoelectron probability. Time-resolved signals according to these expressions are connected to the electronic state \(\Psi(t_{p})\) at the time of measurement. In order to check the applicability of the frozen-density approximation, a situation when the probe-pulse duration is comparable to the characteristic time scale of electron dynamics was considered in Reference [32]. It was assumed that the pump pulse launched coherent electron dynamics by creating an electron hole in a superposition of HOMO, HOMO-1, and HOMO-2 of indole, which was measured by the probe pulse of the same parameters as in the previous section. The shortest beating period of the time-dependent electron density in this case was 1.5 fs, which was still larger than the probe-pulse duration of 1 fs. In order to accurately describe the situation, when the probe pulse duration is comparable with or longer than the characteristic time scale of electron dynamics, one has to take into account the evolution of the electronic system during the action of the probe pulse. The corresponding relation for TRARPES can be derived from the general expression in Equation (40), resulting in \[P^{\text{PE}}(\mathbf{Q}_{\text{el}})= \mathrm{P}_{0}^{\text{PE}}|\epsilon_{\text{in}}\cdot\mathbf{Q}_{ \text{el}}|^{2}\sum_{F,\sigma}\int d^{3}r_{1}\int d^{3}r_{2}e^{i\mathbf{Q}_{ \text{el}}\cdot(\mathbf{r}_{2}-\mathbf{r}_{1})} \tag{51}\] \[\qquad\times\sum_{I,K}\langle\Psi_{K}(t_{p})|\hat{G}_{F}^{\text{ PE}}(\mathbf{r}_{1},\mathbf{r}_{2})|\Psi_{I}(t_{p})\rangle e^{-(\Omega_{FK}^{ \text{el}}-\epsilon_{e})^{2}\tau_{p}^{2}/(8\ln 2)}e^{-(\Omega_{Fl}^{\text{el}}- \epsilon_{e})^{2}\tau_{p}^{2}/(8\ln 2)},\]  
where \(\Psi_{I(K)}(t_{p})=\Phi_{I(K)}e^{-iE_{I(K)}(t_{p}-t_{0})}\) and \(\Omega_{Fl(FK)}^{\text{el}}=\omega_{\text{in}}-E_{F}^{\text{N}_{\text{el}}-1}+ E_{I(K)}\). Please notice that this relation does not depend on the electronic state at the time of measurement \(\mathbf{\Psi}(t_{p})=\sum_{I}C_{I}\Phi_{I}e^{-iE_{I}(t_{p}-t_{0})}\), since the corresponding sum cannot be singled out due to the factors \(e^{-(\Omega_{Fl(FK)}^{\text{el}}-\epsilon_{e})^{2}\tau_{p}^{2}/(8\ln 2)}\). The authors compared angle-averaged photoelectron spectra at four probe-pulse arrival times calculated with Equation (51) taking into account the evolution of the electronic system during the action of the probe pulse shown in Figure 8a and with Equation (42) assuming the frozen-density approximation shown in Figure 8b. The dramatical difference between Figure 8a and Figure 8b demonstrated that the frozen-density approximation broke down and provided completely incorrect results. Thus, they demonstrated that if the probe-pulse duration is comparable with or longer than the shortest beating period of the electron density of a given system, the generated photoelectron spectra are not connected to its instantaneous electronic state as assumed in Equation (42). At the same time, they verified that a probe-pulse duration being ten times shorter than the oscillation period of the electron density is enough to satisfy the frozen-density approximation. Let us consider the consequence of the evolution of the electronic system during the action of the probe pulse on time-resolved scattering patterns obtained by means of NRXS and RXS.