Ultrafast population transfer to excited valence levels of a molecule driven by x-ray pulses

First-principles quantum-mechanical calculations of an intense-field ultrafast two-color core-hole stimulated Raman process in nitric oxide are presented. They employ the multiconfiguration time-dependent Hartree-Fock (MCTDHF) method with all 15 electrons active. These calculations demonstrate a robust excitation localized on an atom through a core-electron stimulated Raman transition, the first step in proposed stimulated x-ray Raman spectroscopy experiments. A total population transfer of approximately 41% into valence excited states and 30% ionization is obtained via two concurrent 1.31-fs pulses with maximum intensities of 0.5 and $3\times {10}^{17}\text{W}{\mathrm{cm}}^{-2}$. It is found that both resonant and nonresonant (via the continuum) Raman transitions contribute. All aspects of these calculations except for ac Stark shifts are converged with a modest basis of 11 orbitals, demonstrating the efficiency of MCTDHF for the treatment of nonperturbative electronic dynamics in molecules.

Figure 1

Schematic of the stimulated Raman transitions coherently populating the $B^{\prime }{}^2\Delta ,G{}^2{\Sigma }^{-}$, and $I{}^2{\Sigma }^{+}$ states of NO.

Figure 2

(top) Fourier transform of the two-color pulse used for the stimulated Raman process; (bottom) real and imaginary components of the expectation value of the instantaneous time-dependent Hamiltonian $H=H_0+\vec{A}\left(t\right)·\nabla$, calculated in the velocity gauge, with 10 orbitals.

Figure 3

Absorption [the quantity $\epsilon {\tilde{S}}^{+}\left(\epsilon \right)$ calculated from Eq. (2), arbitrary units] calculated with only the pump laser for 10 and 11 orbitals, showing large negative ac Stark shift. (left) Absorption calculated at the intensity used in the stimulated Raman calculation, $5\times {10}^{16}\text{W}{\mathrm{cm}}^{-2}$; (right) that calculated at $5\times {10}^{11}\text{W}{\mathrm{cm}}^{-2}$.

Figure 4

Absorption [the quantity $\epsilon {\tilde{S}}^{+}\left(\epsilon \right)$ calculated from Eq. (2), arbitrary units] of the pump laser calculated with both pump and Stokes pulses, varying the strength of the Stokes pulse, for (left) 10 and (right) 11 orbitals. (top) Pump laser at $5\times {10}^{16}\text{W}{\mathrm{cm}}^{-2}$, as in the stimulated Raman calculation. The solid black lines in the top panels correspond to the parameters of the stimulated Raman calculation with the Stokes pulse at full intensity. (bottom) Pump laser at $5\times {10}^{11}\text{W}{\mathrm{cm}}^{-2}$.

Figure 5

Absorption and emission spectra: the unitless function $\epsilon {\tilde{S}}^{+}\left(\epsilon \right)$ calculated from Eq. (2) as obtained through Fourier transforms of the dipole moment in the directions parallel (thick lines) and perpendicular (thin lines) to the bond axis, aligned with the Stokes and pump pulses, respectively. Results are shown for four different orbital bases, and those for the 10-orbital basis are shown in the length and velocity gauges and also with a larger azimuthal basis $\left(m=3\right)$; those lines coincide on the scale of the figure. The integral of the grey curves is the change in energy $(\sim 50$ eV) shown in Fig. 2.

Figure 6

Populations during the pulse for calculations with different numbers and types of initial orbitals. The top two rows show populations of core-hole states. The bottom two rows show populations of valence states. The molecule begins as ${}^2\Pi ,M=+1$, and the populations of the two $M$ components of each of the degenerate $\Pi$ and $\Delta$ states are plotted separately. Populations are, by definition, gauge dependent.