Theoretical study of molecular electronic and rotational coherences by high-order-harmonic generation

The detection of electron motion and electronic wave-packet dynamics is one of the core goals of attosecond science. Recently, choosing the nitric oxide molecule as an example, we have introduced and demonstrated an experimental approach to measure coupled valence electronic and rotational wave packets using high-order-harmonic-generation (HHG) spectroscopy [Kraus et al., Phys. Rev. Lett. 111, 243005 (2013)]. A short outline of the theory to describe the combination of the pump and HHG probe process was published together with an extensive discussion of experimental results [Baykusheva et al., Faraday Discuss. 171, 113 (2014)]. The comparison of theory and experiment showed good agreement on a quantitative level. Here, we present the theory in detail, which is based on a generalized density-matrix approach that describes the pump process and the subsequent probing of the wave packets by a semiclassical quantitative rescattering approach. An in-depth analysis of the different Raman scattering contributions to the creation of the coupled rotational and electronic spin-orbit wave packets is made. We present results for parallel and perpendicular linear polarizations of the pump and probe laser pulses. Furthermore, an analysis of the combined rotational-electronic density matrix in terms of irreducible components is presented that facilitates interpretation of the results.

Figure 1

(a) Illustration of HHG starting from a coherent superposition of two electronic states: the first two pathways, HHG from different electronic channels, happen independently (direct HHG channels); the other two pathways are connecting different electronic states (cross channel HHG). These channels only contribute to a macroscopic signal when they are coherently connected. In the density-matrix description it is the off-diagonal matrix element (the coherence) that determines the HHG signal. (b) Rotational level diagram of the two electronic states $F_1$ and $F_2$ of NO.

Figure 2

$J$ state populations for states $F_1$ and $F_2$ before and after the alignment pulse with the inset of the temporal populations for those two states.

Figure 3

Temporal variations of the multipole coefficients $\mathrm{Re}\left[f_{K,0}^{11}\left(t\right)\right]$ and $\mathrm{Re}\left[f_{K,0}^{22}\left(t\right)\right]$ for $K=0$, 2, 4, and 6, which are related to the electronic occupations.

Figure 4

Temporal variations of the multipole coefficients $\mathrm{Re}\left[f_{K,\pm 2}^{12}\left(t\right)\right]$ for $K=2,4$, and 6, which are related to the electronic coherence.

Figure 5

Temporal contour plots of the quantity $W^{\mathrm{DC}}(\theta ,\beta ,\tau )$ for $\beta =0$ and π/2, which corresponds to parallel and perpendicular polarizations of the pump and probe pulses, respectively.

Figure 6

Temporal contour plots of the quantity $W^{\mathrm{CC}}(\theta ,\beta ,\tau )$ for $\beta =0$ and π/2.

Figure 7

Yield of the ninth harmonic (H9) as a function of pump-probe time delay for $\beta =0$ and the Fourier transform for both theory (upper panel) and experiment (lower panel). Furthermore, we show the expected signals (shifted up) by omitting the coherent cross channel contribution.

Figure 8

Yield of the 15th harmonic (H15) as a function of pump-probe time delay for $\beta =0$ and the Fourier transform for both theory (upper panel) and experiment (lower panel). Furthermore, we show the expected signals (shifted up) by omitting the coherent cross channel contribution.

Figure 9

Yield of harmonics H9 and H15 as a function of pump-probe delay time for parallel $\left(\beta =0\right)$ and perpendicular $(\beta =\pi /2)$ polarization of pump and probe laser fields.

Figure 10

$J$ state populations for states $F_1$ and $F_2$ before and after the alignment pulse. The inset shows the temporal evolution of the total occupation in states $F_1$ and $F_2$. The upper panel shows results for including only the first term (D0) of the interaction Hamiltonian of Eq. (7); the lower panel shows results for treating only the second term (D2).

Figure 11

Temporal evolution of the off-diagonal reduced electronic density-matrix element ${\widehat{\rho }}_{12}^{\mathrm{el}}\left(t\right)$ and the degree of coherence $\mathbb{C}\left(t\right)$ for interactions D0, D2, and D0 + D2 terms of interaction Hamiltonian of Eq. (7).

Figure 12

Temporal evolution of the multipole coefficients $\mathrm{Re}\left[f_{K,0}^{11}\left(t\right)\right]$ and $\mathrm{Re}\left[f_{K,0}^{22}\left(t\right)\right]$ for $K=0$, 2, 4, and 6, and $\mathrm{Re}\left[f_{K,\pm 2}^{12}\left(t\right)\right]$ for $K=2$, 4, and 6 for interactions D0 and D2.

Figure 13

Temporal evolution of the purely geometric quantity $W^{\mathrm{CC}}(\theta ,\beta ,\tau )$ for interaction D2 for $\beta =0$ and $\beta =\pi /2$.

Figure 14

Yield of harmonics H9 and H15 as a function of the pump-probe time delay for the interactions D0 and D2 for $\beta =0$ and π/2.