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 non-interacting 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.

14 pages, 7 figures