Theory for time-resolved resonant inelastic x-ray scattering

Time-resolved measurements of materials provide a wealth of information on quasiparticle dynamics, and have been the focus of optical studies for decades. In this paper, we develop a theory for explicitly evaluating time-resolved resonant inelastic x-ray scattering (tr-RIXS). We apply the theory to a noninteracting electronic system and reveal the particle-hole spectrum and its evolution during the pump pulse. With a high-frequency pump, the frequency and amplitude dependence analysis of the spectra agrees well with the steady-state assumptions and Floquet excitations. When the pump frequency is low, the spectrum extracts real-time dynamics of the particle-hole continuum in momentum space. These results verify the correctness of our theory and demonstrate the breadth of physical problems that tr-RIXS could shed light on.

Figure 1

An illustration of the RIXS process. It has two steps: In the excitation step, the incoming photon with energy ${\omega }_{\mathrm{i}}$ and momentum ${\mathit{k}}_{\mathrm{i}}$ promotes an inner shell electron to the outer shell at site $m$ in the initial state and creates the intermediate state. Later, in the deexcitation step, the intermediate state emits the outgoing photon with energy ${\omega }_{\mathrm{f}}$ and momentum ${\mathit{k}}_{\mathrm{f}}$ and the core hole is filled, leaving the final state with an excitation.

Figure 2

The Keldysh contour of tr-RIXS. The red curve and cyan shade represent the pump and probe pulse, respectively. $t_1,t_2,t_2^{\prime }$, and $t_1^{\prime }$ are as defined in Eq. (9). The probe centered at $t$ has a width $2{\sigma }_{\mathrm{pr}}$, containing all four time points. The separations between $t_1\left(t_1^{\prime }\right)$ and $t_2\left(t_2^{\prime }\right)$ correspond to the time when the core hole is present and are further constrained by $l(t_1,t_2)$ or $l(t_1^{\prime },t_2^{\prime })$, as defined in the main text.

Figure 3

Tr-RIXS spectra for a large pump frequency. (a) Schematic of the pump pulse. The Gaussian profile is shown in blue dashed lines, and the five orange dots correspond to the times when spectra in (b)–(f) are taken. (b)–(f) The tr-RIXS spectra at $t=-2{\sigma }_{\mathrm{pu}},-{\sigma }_{\mathrm{pu}},0,{\sigma }_{\mathrm{pu}},2{\sigma }_{\mathrm{pu}}$, respectively. The inset of (b) shows the Fermi surface of the 2D electronic system with red lines, and the blue arrows show the nesting momentum wave vector at $q\simeq 0.4\pi$. Unless otherwise specified, all RIXS intensities are shown in the same arbitrary units.

Figure 4

(a), (b) Tr-RIXS spectra for different pump amplitudes $A_0$ at the pump center: (a) $A_0=0.8$ and (b) $A_0=1.6$. Both pumps lie along $x=y$ with frequency $\mathrm{\Omega }=0.75\mathrm{eV}$. The dashed lines are the Floquet theoretical upper bounds of branches of the particle-hole continuum. The color scale is the same as in Fig. 3. (c) The autocorrelation $\mathcal{C}(\mathit{q},\eta ,t)$ for $\mathit{q}=(\pi ,0)$ at $t=0$. The black, magenta, and green lines show the theoretical first, second, and third Floquet frequencies $n\mathrm{\Omega }$, respectively. $A_0$ is kept at 2.4 throughout.

Figure 5

Tr-RIXS spectra of the 1D system with a low pump frequency. (a) Schematic picture of the pump pulse. The Gaussian profile is shown in blue dashed lines, and the four orange dots correspond to the time when spectra in (b)–(e) are taken. (b)–(e) The tr-RIXS spectra at $t=-720{\mathrm{eV}}^{-1},-150{\mathrm{eV}}^{-1},300{\mathrm{eV}}^{-1},720{\mathrm{eV}}^{-1}$, respectively. The insets plot the band structure and electron distributions in momentum space (shown in red) at each time point.

Figure 6

Comparison of (a) the negative imaginary part of the Lindhard response calculated from Eq. (A3) and (b) the equilibrium RIXS spectrum $I({\omega }_{\mathrm{i}},\mathrm{\Delta }\omega ,\mathit{q})$ calculated from Eq. (A2) for the 2D tight-binding model Eq. (17) as parametrized in Sec. 3a. In numerical evaluation, $\mathrm{\Gamma }=0.80\mathrm{eV}$ and $\delta =0.042\mathrm{eV}$. $\mathit{q}=(q,0)$ for both calculations.

Figure 7

The middle part (i.e., $n$ close to 0) of the Floquet band structure calculated from Eq. (B3) by only considering ${\mathcal{J}}_0$ and considering up to ${\mathcal{J}}_4$. In evaluating Eq. (B3) $r$ ranges from $-12$ to 12. Since the Fourier transform is linear, here we only considered terms in Eq. (B5) related to the $x$ direction (i.e., essentially a 1D tight-binding model). The two calculations are almost indistinguishable.