## 3 Results and Discussions
We finally show the effect of the helicity of the circularly polarized probe pulse on the photoelectron emission. Figure 5 displays results of photoelectron momentum \(\mathcal{J}(p_{x},p_{y})\) and angular \(\mathcal{J}(\theta)\) distributions with right handed helicity \(\xi=1\). The other laser parameters are the same as those  
\[\psi_{\pi_{u}}(\mathbf{r})=[\psi_{\pi_{u}}^{+}(\mathbf{r})+\psi_{\pi_{u}}^{-} (\mathbf{r})]/\sqrt{2}. \tag{8}\]1 2D Photoelectron momentum \(\mathcal{J}(p_{x},p_{y})\) distributions as a function of the time delay \(\Delta\tau\) between the linearly \(x\) polarized \(\lambda_{pu}=70\) nm pump pulse and the circularly \((x,y)\) polarized \(\lambda_{pr}=5\) nm probe pulse with left-handed helicity \(\xi=-1\). The time delay \(\Delta\tau\) varies: **(a)**\(\Delta\tau=4.75\tau_{pu}\); **(b)**\(5.0\tau_{pu}\); **(c)**\(5.25\tau_{pu}\); **(d)**\(5.5\tau_{pu}\); **(e)**\(5.75\tau_{pu}\), and **(f)**\(6.0\tau_{pu}\). The molecule H\({}_{2}^{+}\) is aligned along the \(z\)-axis. The pulse intensities \(I_{0}=1.0\times 10^{14}\) W/cm\({}^{2}\) (\(E_{0}=5.34\times 10^{-2}\) a.u.) and duration \(T_{pu}=10\tau_{pu}=2.32\) fs (1.16 fs FWHM) and \(T_{pr}=10\tau_{pr}=165.4\) as (83 as FWHM) are always fixed. Inserts (yellow curves) are photoelectron angular \(\mathcal{J}(\theta)\) distributions obtained from Equation (6). used in \[\psi_{\pi_{u}}(\mathbf{r})=[\psi_{\pi_{u}}^{+}(\mathbf{r})+\psi_{\pi_{u}}^{-} (\mathbf{r})]/\sqrt{2}. \tag{8}\]2 Figure 5 shows that similar phenomena are produced, except with a clock-wise rotation. Asymmetric photoelectron angular and momentum distributions resulting from the coherent excitation by the \(\lambda_{pu}=70\) nm pump pulse are produced as well. Varying the time delay \(\Delta\tau\) gives rise to a periodical rotation of photoelectron distributions with period 232 as, as predicted in Equation (10). It is found that the rotation with a clock-wise direction follows the right handed helicity of the circularly polarized probe pulse. The combination of Figures 4 and 5 shows that the evolution of the probing photoelectron angular and momentum distributions with time arises from the coherent excitation between the ground and excited electronic state by the pump pulse while the rotation corresponds to the helicity of the circularly polarized attosecond probe pulse. Comparison of results in Figures 3-5 shows that the time-resolved photoelectron angular and momentum distributions depend on the polarization and helicity of the probe pulse. By a linearly polarized pump pulse, the coherent electronic state is degenerate with \(m=\pm 1\). The subsequent ionization from the coherent superposition state are dependent on the properties of the probe pulse. The evolution of the probing photoelectron angular and momentum distributions with the time delay depends on the polarization and helicity of the soft X-ray probe pulse, as illustrated in Figures 3-5. This differs from the coherent excitation process by a circularly polarized pump pulse, in which the excited \(2p\pi_{u}\) state is non-degenerate, with \(m=1\) or \(-1\). As we reported previously [43], the rotation of time-delay dependent photoelectron distributions is determined by the pump pulse. Altering the helicity of the probe pulse cannot vary the rotation of the time-resolved photoelectron distribution. The dependence of the time-resolved photoelectron distributions on the polarization and helicity of the soft X-ray probe pulse indicates the permanent property of the coherent superposition state in the coherent excitation and charge migration. Of interest is that, in the present work, we consider the process of molecular resonant excitations with \(\sigma_{g}-\pi_{u}\) symmetries in the molecular ion H\({}_{2}^{+}\).