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our calculations, we consider cuts along k$_{x}$ -axis in the Brillouin zone at a k$_{y}$ chosen near the zone boundary. Fig. 3 shows the snapshots of the image of the spec- tral function A ( k , ω ) as a function of k$_{x}$ and ω . From Fig. 3(a$_{1}$,a$_{2}$), we observe kinks at ω = − (Δ$_{0}$( T$_{e}$ ) + Ω$_{0}$) (red lines) and − ( E$_{M}$ + Ω$_{0}$) (black lines) at τ = − 100 and − 2 . 7. The red line is shifted in energy as time elapses from τ = − 100 to τ = − 2 . 7 because the BCS gap Δ$_{0}$( T$_{e}$ ) drops significantly from 0.19 to 0.1. These kinks cor- respond to the dip structures discussed in Fig. 2. As τ > − 2 . 3, the superconducting gap vanishes, so do the kinks marked with red lines (see Fig. 3(a$_{3}$-a$_{4}$)). The kinks marked with black lines persist through the whole time sequence, which can be ascribed to the el-ph coupling signature in the normal state. Therefore we have ev- ery reason to regard these kinks marked with red lines as the signature of the el-ph coupling in the supercon- 

ducting state. For hot phonon pumping (Fig. 3(b$_{1}$-b$_{4}$)), in addition to those signatures of the el-ph coupling observed for electron pumping, new kink structure at − (Ω$_{0}$ − Δ$_{0}$( T$_{e}$ ))(marked with magenta line in Fig. 3(b$_{2}$) is also observed. These observations are consistent with our finding in the DOS. In the viewpoint that the elec- tronic spin fluctuations arise from the strong correlation within the electronic DoF itself, and if one can assume that the spin fluctuations will ride on the electrons and thus its effective temperature will be tied to the electronic temperature, direct pumping of hot phonons will provide a unique way to differentiating the bosonic modes be- ing of electronic or phononic origin, to which electronic quasiparticles are strongly coupled in HTSC. 

In conclusion, we have developed a three-temperature model to simulate the time evolution of temperature for electrons, hot phonons, and the lattice. Our model takes both quasiparticle excitation and relaxation into account and thus is valid in both normal and superconducting states, because at the early stage of excitation, the sys- tem can be still in the supercondcting state. Based on this model, we derive a formula for the time-resolved spectral function, allowing us to investigate the dynamics of the el-ph coupling for HTSC materials. 

Acknowledgments. We thank A. V. Balatsky, El- bert E. M. Chia, Hari Dahal, J. K. Freericks, M. Graf, A. Piryatinski, A. J. Taylor, S. A. Trugman, and D. Yarotski for valuable discussions. This work was sup- ported by the National Nuclear Security Administration of the U.S. DOE at LANL under Contract No. DE- AC52-06NA25396, the U.S. DOE Office of Science, and the LDRD Program at LANL. 

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