## 3 Results and Discussions
Because the oscillation period of the coherent electron is very short, \(\tau=232\) as, a soft X-ray attosecond probe pulse is necessary. We use a \(\lambda_{pr}=5\) nm (\(\omega_{pr}=9.11\) a.u.) and \(T_{pr}=10\tau_{pr}\) (83 as FWHM) probe pulse to ionize the excited molecule prepared by the linearly \(x\) polarized \(\lambda_{pu}=70\) nm pump pulse in Figure 2. The use of the soft X-ray pulse has the following advantage in probing the electronic coherence. First the high frequency single photon process can avoid additional excitation effects resulting from other lower electronic states. Second, the extremely short duration of the X-ray pulse on the attosecond time scale can be utilized to resolve the coherent electron rotation with the period of a few hundred attoseconds. Figure 3 displays molecular photoelectron momentum \(\mathcal{J}(p_{x},p_{y})\) and angular \(\mathcal{J}(\theta)\) distributions produced by a linearly polarized \(\lambda_{pr}=5\) nm attosecond probe pulse with its field vector along the \(x\)-axis, i.e., \(\xi=0\). The time delay \(\Delta\tau\) between the pump and probe pulses varies from \(\Delta\tau=4.75\tau_{pu}\) to \(6.0\tau_{pu}\) with a time interval of \(0.25\tau_{pu}\). The other laser parameters are always fixed at pulse intensities \(I_{0}=1.0\times 10^{14}\) W/cm\({}^{2}\) (\(E_{0}=5.34\times 10^{-2}\) a.u.) and durations \(T_{pu}=10\tau_{pu}=2.32\) fs (1.16 fs FWHM) and \(T_{pr}=10\tau_{pr}=165.4\) as (83 as FWHM). The proper intensity of the probe pulse should be chosen. With lower intensities of the X-ray pulse, the probing photoelectron signal will be weak due to small transition dipole moments,whereas, with very strong intensities, the broad spectral width of the X-ray pulse will induce additional unexpected strong field phenomena. Since we focus on the probing photoelectron emission produced by the soft X-ray pulse, only the photoelectron distributions around the momentum \(p_{e}=4.01\) a.u., where \(p_{e}=\pm\sqrt{2(\omega_{pr}-I_{p})}\) corresponds to one \(\omega_{pr}\) photon absorption, are presented in Figure 3. The angular distributions calculated in Equation (6) are also displayed. Figure 3 shows that asymmetric photoelectron angular and momentum distributions are produced, which are shown to be dependent on the time delay \(\Delta\tau\). Altering the time delay \(\Delta\tau\) results in a change of the asymmetry. At \(\Delta\tau=4.75\tau_{pu}\), the distributions in the right half angular and momentum plane dominate. Increasing \(\Delta\tau\) leads to a decrease of the right momentum ring and an increase of the left in amplitude. At \(\Delta\tau=5.0\tau_{pu}\) the amplitudes of the momentum distributions in the left and half planes are comparable, and at \(\Delta\tau=5.25\tau_{pu}\) the left one reaches the maximum value. Increasing the time delay \(\Delta\tau\) further, a reverse process occurs where the left distribution decreases while the right one increases. At \(\Delta\tau=5.5\tau_{pu}\), the comparable distributions in the left and right planes are obtained, and at \(\Delta\tau=5.75\tau_{pu}\), the right one dominates again. The periodical evolution of the photoelectron momentum distributions with the time delay \(\Delta\tau\) illustrates the electronic coherent in the resonant excitation by the pump pulse. The evolution period is \(\tau=\tau_{pu}=2\pi/\Delta E\), the same as the coherent electron wave packets in Equation (10) and Figure 2. The electronic coherence gives rise to asymmetric electron density distributions of the coherent superposition state \(\psi_{0}(\mathbf{r},t)\). Figure 2: Evolution of the electron density probabilities \(\mathcal{P}(x,y)\) with the time \(t\) in the molecular ion H\({}_{2}^{+}\) aligned along the \(z\)-axis excited by a linearly polarized \(\lambda=70\) nm pump pulse with its field vector along the \(x\)-axis at intensity \(I_{0}=1.0\times 10^{14}\) W/cm\({}^{2}\) (\(E_{0}=5.34\times 10^{-2}\) a.u.) and duration \(T=10\tau_{pu}=2.32\) fs.