Time-resolved resonant inelastic x-ray scattering in a pumped Mott insulator

Collective excitations contain rich information about photoinduced transient states in correlated systems. In a Mott insulator, charge degrees of freedom are frozen, but can be activated by photodoping. The energy-momentum distribution of the charge excitation spectrum reflects the propagation of charge degrees of freedom and provides information about the interplay among various intertwined instabilities on the timescale set by the pump. To reveal charge excitations out of equilibrium, we simulate time-resolved x-ray absorption and resonant inelastic x-ray scattering using a Hubbard model. After pumping, the former resolves photodoping, while the latter characterizes the formation, dispersion, weight, and nonlinear effects of collective excitations. Intermediate-state information from time-resolved resonant inelastic x-ray scattering (trRIXS) can be used to decipher the origin of these excitations, including bimagnons, Mott-gap excitations, doublon and single-electron in-gap states, and anti-Stokes relaxation during an ultrafast pump. This paper provides a theoretical foundation for existing and future trRIXS experiments.

Figure 1

Schematic illustrating the pump-probe processes of indirect trXAS and trRIXS. Left to right: The pump field drives the electronic system out of equilibrium. After a certain time delay, the probe photon with a frequency ${\omega }_{\mathrm{in}}$ (resonant to an x-ray edge) excites a core-level electron into a high unoccupied level, leading to the trXAS spectrum. The core-hole attraction then generates collective excitations of the valence electrons. Finally, the photoelectron and core hole annihilate, emitting a photon with a frequency of ${\omega }_{\mathrm{s}}$. This photon-in-photon-out process gives the trRIXS spectrum, which characterizes the remaining final state of the valence electrons, carrying collective excitations with energy ${\omega }_{\mathrm{in}}-{\omega }_{\mathrm{s}}$.

Figure 2

(a) Snapshots of trXAS spectra at $t=-6t_h^{-1},-3t_h^{-1}$, 0, $3t_h^{-1}$, and $6t_h^{-1}$. Different peaks are marked by different colors, corresponding to the processes in Fig. 3. (b) False-color plot of the trXAS spectra during and after the pump. The pump pulse is indicated by the white curve and the times of the snapshots in panel (a) are marked by red circles. The colored arrows denote the different processes of panel (a).

Figure 3

(a) Upper to lower sprectrum: Postpump trXAS $\left(t=10t_h^{-1}\right)$ for half-filled system in Fig. 2, equilibrium XAS for the same half-filled system, equilibrium XAS for 16.7% hole-doped and 16.7% electron-doped systems, respectively. (b), (c) Schematic of four types of resonant absorption processes with intermediate states: (b) the “poorly screened” and “well-screened” channels present at half-filled Mott insulators; (c) processes corresponding to double occupancy (doublon) and empty site (hole). The four resonances are denoted by different colors in (a), following the color code of (b), (c).

Figure 4

The $\mathbf{q}=(\pi ,\pi /3)$ trRIXS spectra during a $\mathrm{\Omega }=5t_h$ pump in (a1)–(a6) energy loss-incident energy $\left(\omega -{\omega }_{\mathrm{in}}\right)$ and (b1)–(b6) scattering energy-incident energy $\left({\omega }_{\mathrm{s}}-{\omega }_{\mathrm{in}}\right)$ view. The four XAS resonances ${\omega }_{\mathrm{in}}-E_{\mathrm{edge}}$= $-21.5,-16,-10.5$, and 2.5 $t_h$ are labeled as green, blue, red, and yellow arrows, respectively, corresponding to the double-occupancy, poorly screened, well-screened, and empty-site intermediate states. The arrows to the right denote the final-state resonances of ${\omega }_{\mathrm{s}}=-1.32$ (yellow), $-12.27t_h$ (red), $-17.62t_h$ (blue), and $-22.53t_h$ (green). The white dashed lines denote the elastic response, while the colored lines in (a6) and (b6) represent the resonance for initial and final states. The upper insets show the equilibrium XAS spectrum and the corresponding time during a pump pulse.

Figure 5

Schematic illustrating the two significant excitations revealed by trRIXS: (a) the Mott-gap excitation through the “well-screened” initial state and “double-occupancy” final state; (b) the bimagnon excitation through the “poorly-screened” initial state and “poorly screened” final state, and through the “well-screened” initial state and “well-screened” final state.

Figure 6

(a) Time evolution of the dynamical charge structure factor $N(\mathbf{q},\omega ,t)$ in the same system of Fig. 4 and pump conditions for $\mathbf{q}=(\pi ,\pi /3)$. The inset denotes the evolution of pump field and the right-hand side markers guide the eye for the Mott-gap excitations and photoinduced excitations. (b)–(e) The evolution of trRIXS spectrum for four initial-state resonances: (b) poorly screened, (c) well-screened, (d) double-occupancy, and (d) empty-site channels.

Figure 7

Time evolution of the dynamical charge structure factor $N(\mathbf{q},\omega ,t)$, trRIXS snapshots before and after the pump, and the time evolution of trRIXS spectra at four resonant ${\omega }_{\mathrm{in}}\mathrm{s}$ for (a) $\mathbf{q}=(\pi /2,\pi /2)$ and (b) $\mathbf{q}=(\pi ,\pi )$. The pump condition and the layout are the same as Fig. 6.

Figure 8

(a) Three snapshots of the $\mathbf{q}=(\pi ,\pi /3)$ trRIXS spectra with the same pump and probe condition as Fig. 4 but a longer core-hole lifetime of ${\tau }_{\mathrm{core}}=t_h^{-1}$. The white dashed lines denote the elastic response, while the colored lines represent the resonance for initial and final states in the same manner as Fig. 4. (b) Time evolution of the trRIXS spectra system for two initial-state resonances: (b) poorly screened channel and (c) well-screened channel.

Figure 9

(a) Three snapshots of the $\mathbf{q}=(\pi ,\pi /3)$ trRIXS spectra with the same pump condition as Fig. 4 but a shorter probe width of ${\sigma }_{\mathrm{pr}}=0.5t_h^{-1}$. The white dashed lines denote the elastic response. (b) Time evolution of the trRIXS spectra system for two initial-state resonances: (b) poorly screened channel and (c) well-screened channel.

Figure 10

Time evolution of the dynamical charge structure factor $N(\mathbf{q},\omega ,t)$, trRIXS snapshots before and after the pump, and the time evolution of trRIXS spectra at four resonant ${\omega }_{\mathrm{in}}\mathrm{s}$ for (a) $\mathbf{q}=(0,2\pi /3)$ and (b) $\mathbf{q}=(\pi /2,\pi /6)$. The pump condition and the layout are the same as Fig. 6.

Figure 11

Time evolution of the dynamical charge structure factor $N(\mathbf{q},\omega ,t)$, trRIXS snapshots before and after the pump, and the time evolution of trRIXS spectra at four resonant ${\omega }_{\mathrm{in}}\mathrm{s}$ for (a) $\mathbf{q}=(0,0)$ and (b) $\mathbf{q}=(\pi /2,6\pi /6)$. The pump condition and the layout are the same as Fig. 6.