Experimental signatures of phase interference and subfemtosecond time dynamics on the incident energy axis of resonant inelastic x-ray scattering

Core hole resonance is used in x-ray spectroscopy to incisively probe the local electronic states of many-body systems. Here, resonant inelastic x-ray scattering (RIXS) is studied as a function of incident photon energy on Mott insulators ${\mathrm{SrCuO}}_2$ and NiO to examine how resonance states decay into different excitation symmetries at the transition-metal $M$, $L$, and $K$ edges. Quantum interference patterns characteristic of the two major RIXS mechanisms are identified within the data, and used to distinguish the attosecond scale scattering dynamics by which fundamental excitations of a many-body system are created. A function is proposed to experimentally evaluate whether a particular excitation has constructive or destructive interference in the RIXS cross section, and corroborates other evidence that an anomalous excitation is present at the leading edge of the Mott gap in quasi-one-dimensional ${\mathrm{SrCuO}}_2$.

Figure 1

Overlapping core hole resonance states. Top: Inelastic fluorescence yield at the Cu $M$ edge of ${\mathrm{SrCuO}}_2$ is fitted by two overlapping Lorentzians with a 2:1 intensity ratio. Bottom: (blue) RIXS spectra measured at incident energies corresponding to the $x$ axis of the top panel reveal three energy-loss peaks representing $d-d$ excitations. Red curves show an atomic multiplet simulation.

Figure 2

Two-slit RIXS quantum interference patterns. The orbital transitions involved in creating a (a) $3d_{3z^2-r^2}$ or (b) $3d_{xy}$ excitation through $M_3$ and $M_2$ core hole resonances are shown. Gray lines: Constant phase contours are shown for the scattering process in which (c) $3d_{3z^2-r^2}$ and (d) $3d_{xy}$ orbiton excitations are created in ${\mathrm{SrCuO}}_2$. The constant phase contours from scattering through $M_3$ and $M_2$ resonance are offset in time by (black arrows) the phase difference between ${\Theta }_{M2}$ and ${\Theta }_{M3}$, and interfere constructively when they converge. (e),(f) The resulting interference patterns are seen from (circles) incident energy dependence of the $3d_{3z^2-r^2}$ and $3d_{xy}$ RIXS intensity, and (solid line) fitted using Eq. (2) with phase shifts ${\theta }_{M3,z^2}={\theta }_{M2,z^2}+\pi$ and ${\theta }_{M3,xy}={\theta }_{M2,xy}$. A dashed line shows the fit result with quantum interference disregarded by moving the sum over intermediate states (${\sum }_m$) outside of the absolute-value brackets in Eqs. (1) and (2). Arrows highlight the change from quantum interference.

Figure 3

Quantum interference in the experimental RIXS profile of NiO. (a) A simulation of RIXS from NiO is displayed with a red-hot color scale. Energy regions in which intensity is dominated by different inelastic features are numbered 1–4. (b) Red curve: An estimate of quantum interference at $h\nu =65.25\text{eV}$ [indicated by a vertical line in panels (a) and (c)] in the simulated RIXS profile is obtained from the $\zeta (E,h\nu )$ function [Eq. (C3)], and compared with (dark green curve) the exact fractional effect of quantum interference on scattering intensity, as defined in the text. Red points: Experimental values of $\zeta (E,h\nu )$ are shown with error bars, where adequate statistics can be obtained. (c) Very different patterns are seen in the incident energy dependence of inelastic fluorescence yield (FY), and RIXS in the four energy-loss regions numbered in panel (a). The incident energy at which FY onset has a maximal slope ($h\nu =65.25\text{eV}$) is identified with a red stripe.

Figure 4

Quantum interference at the copper $K$ edge. (a) The incident energy dependence of $K$-edge RIXS is shown for SCO. (b) (black) RIXS curves and (red) the $\zeta (E,h\nu )$ function measured at the leading edge of resonance ($h\nu =8983$ eV) are shown for the center and boundary of the one-dimensional (1D) single-chain Brillouin zone. Shaded energy regions are associated with a pregap feature, Mott gap excitations, and metal-ligand charge transfer (CT) excitations, respectively. (c) Top: X-ray absorption is estimated from fluorescence yield, with incident polarization matching the RIXS measurements. Bottom: (black) Incident energy dependence of RIXS intensity in the 3–4-eV Mott gap energy-loss window is compared with two-peak fits that assume (red) fully destructive and (blue) fully constructive path phases. Circles mark points on the red and black curves at which intensity has dropped by $50\%$ relative to the nearest local maximum.

Figure 5

Predicting interference at the Ni $L$ edge. (a) The incident energy dependence of $L$-edge RIXS is modeled for NiO. (b) Quantum interference at $L_3$ (852 eV) and $L_2$ (869.4 eV) is estimated from (red curve) the $\zeta (E,h\nu )$ function [Eq. (C3)], and compared with (dark green curve) the exact fractional effect of quantum interference on scattering intensity. Regions at which the $\zeta (E,h\nu )$ curve would be easiest to measure experimentally with good statistics are highlighted in red.

Figure 6

Transforming the RIXS process into the time domain. (a) Fractional intensity of the $3d_{xy}$ orbiton and $3d_{3z^2-r^2}$ spin and orbital excitation in the RIXS profile is simulated as a function of core hole lifetime, using Eq. (2) with experimental fit parameters. The expected trend for constructive and destructive path phases is indicated with red and black arrows, respectively. (b) Scattering intensity of several significant final states of NiO is simulated as a function of core hole lifetime using an AM calculation. Curves represent the scattering intensity for (red) elastic scattering, (black) spin rotation excitations near 0.1–0.2 eV, and (green) the $\sim 1$-eV $t_{2g}$ to $e_g$ orbital excitation (${}^3{T}_2$ excitation manifold). Time scales of $M$-edge core hole lifetime are indicated with blue shaded regions, and dashed vertical lines indicate time scales associated with core hole energetic parameters. The normalizing factor $R_{\mathrm{tot}}$ is defined as the sum of $3d_{3z^2-r^2}$ and $3d_{xy}$ excitation intensity for panel (a), and the sum over all atomic multiplet excitations for panel (b).

Figure 7

Quantum interference above the band gap. (a) A simulation of RIXS from NiO is displayed with a red-hot color scale. Intensity above the insulating band gap is multiplied by 10 for better visibility. (b) Red curve: The $\zeta (E,h\nu )$ function defined in the main text is used to estimate quantum interference at $h\nu =65.25\text{eV}$ in the calculated RIXS profile. Dark blue curve: The exact fractional amplitude of quantum interference is evaluated from the AM calculation, as described in the main text.