Nonlinear light scattering in molecules triggered by an impulsive X-ray Raman process

Abstract

The time-and-frequency resolved nonlinear light scattering (NLS) signals from a time evolving charge distribution of valence electrons prepared by impulsive X-ray pulses are calculated using a superoperator Green's function formalism. The signal consists of a coherent ~ N²-scaling difference frequency generation and an incoherent fluorescence ~ N-scaling component where N is the number of active molecules. The former is given by the classical Larmor formula based on the time-dependent charge density. The latter requires additional information about the electronic structure and may be recast in terms of transition amplitudes representing quantum matter pathways.

FIG. 1

(Color online) (a) - setup for the X-ray induced NLS, (b) - level scheme of the model system, g₀,..., g₁ represent valence states, f are core excited states.

FIG. 2

(Color online) (a) loop diagram for the bare incoherent signal in a gated measurement caused by a single molecule α, (b) coherent signal generated by a pair of molecules α ≠ β. Eqs. (2) - (3) (coherent) and Eqs. (8) - (9) (incoherent) can be read o these diagrams (for diagram rules see Ref. [38]). The shaded area represents an excitation by an arbitrary sequence of pulses, which prepares the molecule in a superposition state.

FIG. 3

Various valence level-schemes considered in this paper. (a) - a general level scheme with arbitrary transition dipoles. (b) - a two band model with only interband transition dipole, (c) - two bands with a single excited state. (d) - two bands with a single ground state.

FIG. 4

(Color online) The stimulated X-ray Raman induced NLS process. Straight arrows correspond to interactions with X-ray pulses, wavy arrows represent spontaneous emission. (a) loop diagram for the bare SRIF signal (b) the bare coherent DFG signal, (c) The level scheme used for cysteine is composed of a ground state g₀ and 50 valence excited states g₁ with energies between ~5.75eV and ~11.5eV, f are core excited states.

FIG. 5

(Color online) SRIF (Top) and DFG (Bottom) NLS signals of cysteine following stimulated Raman excitation by an X-ray pulse of width 14.2 eV and central frequency tuned to the nitrogen (~ 404.4 eV), oxygen (~ 532.2 eV), or sulfur (~ 2473.5 eV) K-edges, as indicated. The two signals coincide for g_1,2 ↔ g₀ transitions. Transitions between excited states g_1,2 ↔ g₃, $\stackrel{‒}{\omega }>5\mathrm{eV}$ do not show up in the coherent signal. Γ ~0.04 eV is used for the gating bandwidth.

FIG. 6

(Color online) Time resolved SRIF - solid red line and DFG - dotted blue line signals are compared for X-ray pulses resonant with S, O and N in cysteine. Γ ~0.04 eV is used for the gating bandwidth.

FIG. 7

(Color online) Time and frequency gated SRIF signals of cysteine (Eq. (1)). The gating parameters in atomic units σ_T = 1000, σ_ω = 0.001 - (a), σ_T = 100, σ_ω = 0.01 - (b), and σ_T = 2000, σ_ω = 0.005 - (c). These permit all transitions within ~ 0.054 eV, 0.54eV and 0.1 eV of each other to interfere, respectively. Top panel has optimal gating parameters and reveals both areas of low transition density (where the intensities are time independent) and areas of higher densities (where beats develop as a result of interference between transitions). Note the particularly prominent beating near 9 eV with an approximate period of 120 fs. The actual distance between these states is ~ 0.0126 Ha well within the allowed detection bandwidth. Middle panel has low frequency resolution but rather high temporal resolution that results in clear beating signal. The bottom panel has high spectral but low temporal resolution, that results in the suppression of formerly prominent beating at ~ 9 eV. Further narrowing the frequency-domain gating widths eliminates it altogether.

FIG. 8

(Color online) Frequency-resolved SRIF - green (light gray) line and DFG - red (dark gray) line following a stimulated Raman excitation by two pulses of varying width. In all three panels, the first pulse is tuned to the transition from the bottom of the nitrogen core band to the ground state (~ 388 eV) and has a width of 1.36 eV. The second pulse is tuned to the transition from the bottom of the nitrogen core band to the top of the valence band (~ 374 eV). The second pulse is gradually broadened to allow for more valence excitations prior to the NLS ~1.36 eV (bottom), ~3eV (middle), and ~6 eV (top). Γ ~0.04 used for the gating bandwidth.