Imaging electron-density fluctuations by multidimensional X-ray photon-coincidence diffraction

Abstract

The ultrafast spontaneous electron-density fluctuation dynamics in molecules is studied theoretically by off-resonant multiple X-ray diffraction events. The time- and wavevector-resolved photon-coincidence signals give an image of electron-density fluctuations expressed through the four-point correlation function of the charge density in momentum space. A Fourier transform of the signal provides a real-space image of the multipoint charge-density correlation functions, which reveal snapshots of the evolving electron density in between the diffraction events. The proposed technique is illustrated by ab initio simulations of the momentum- and real-space inelastic scattering signals from a linear cyanotetracetylene molecule.

Keywords: X-ray diffraction; multidimensional spectroscopy; photon-coincidence.

Fig. 1.

(A) Loop diagram for off-resonant X-ray scattering from a single molecule detected by two-photon coincidence. The molecule is initially in the ground state $g$. $T$ is the time delay between the two scattering events. (B) Schematic of the two-photon coincidence diffraction process. (C) Chemical structure of cyanotetracetylene (${\mathrm{H}\mathrm{C}}_{\mathrm{8}}\mathrm{C}\mathrm{N}$) oriented in laboratory frame along $x$ with the ground-state charge density ${\sigma }_{gg}$ (isovalue of 0.05). The color scheme for the atoms is carbon (gray), hydrogen (white), and nitrogen (blue). (D) Energy levels of the ground ($g$) and excited ($e=1,\dots ,10$) states.

Fig. 2.

Off-resonant scattering of a Gaussian X-ray pulse (Eq. 7) from ${\mathrm{H}\mathrm{C}}_{\mathrm{8}}\mathrm{C}\mathrm{N}$. (A) The pump spectral envelope $A_{p1}\left(\omega \right)$ and the detected scattered frequency ${\omega }_{s1}={\mathrm{\Omega }}_p-3.93\hspace{0.17em}\mathrm{e}\mathrm{V}$ (red vertical line). (B) The ground-state ${\mathbf{q}}_1$ scattering pattern created by the first pulse propagating along $y$.

Fig. 3.

Time-dependent off-diagonal elements ${\rho }_{ab}\left(T\right)$ that dominate the dynamics of the signals for the two ${\mathbf{q}}_1$ points A and B marked in Fig. 2B.

Fig. 4.

Two-photon coincidence ${\mathbf{q}}_2$ scattering patterns for the two points marked in Fig. 2B at three time delays $T$. Left shows the full signal $S\left(T=0\right)$. Center and Right show the signal difference $S\left(T\right)-S\left(T=0\right)$.

Fig. 5.

The real (A) and imaginary (B) parts of the real-space two-photon coincidence signal $S\left({\mathbf{q}}_1,{\mathbf{R}}_2,T\right)$ for ${\mathbf{q}}_1$ points A and B marked in Fig. 2B at three time delays $T$. To highlight the changes, A, Center and Right show the signal differences $\mathrm{R}\mathrm{e}\left[S\left({\mathbf{q}}_1,{\mathbf{R}}_2,T\right)-S\left({\mathbf{q}}_1,{\mathbf{R}}_2,T=0\right)\right]$.