Theoretical Description of Time-Resolved Photoemission Spectroscopy: Application to Pump-Probe Experiments

The theory for time-resolved, pump-probe, photoemission spectroscopy and other pump-probe experiments is developed. The formal development is completely general, incorporating all of the nonequilibrium effects of the pump pulse and the finite time width of the probe pulse, and including possibilities for taking into account band structure and matrix element effects, surface states, and the interaction of the photoexcited electrons with the system leading to corrections to the sudden approximation. We also illustrate the effects of windowing that arise from the finite width of the probe pulse in a simple model system by assuming the quasiequilibrium approximation.

Figure 1

Photoemission spectra at half-filling with $U=3$. The temperatures are $T=0.00539$ for panel (a) and $T=0.0189$ for panel (b). The continuous beam photoemission curve is in black and is labeled by $G^{<}(\omega )$, while the pump-probe curves for different probe widths are in other colors and labeled by their widths $\Gamma$. In panel (a), symbols are used to differentiate the different curves. In panel (b), the curves with $\Gamma =10$ and the lesser Green’s function are essentially indistinguishable except for some kinks near $\omega =0$. The $\Gamma =5$ curve has a slightly lower peak, but then merges with the others, while the $\Gamma =2$ curve is clearly distinguished.

Figure 2

Time-resolved photoemission spectra at half filling with $U=3$ ($\Gamma =5$) at temperatures $T=0.00539$ (dashed line) versus the equilibrium continuous beam PES at different $T$s. Note how the windowing effect is different from thermal broadening effects, especially for positive energies.