Theoretical investigation of slow gain recovery of quantum cascade lasers observed in pump-probe experiment

Time-resolved spectroscopy-based pump-probe experiments performed on quantum cascade lasers (QCLs) exhibit an initial fast gain recovery followed by a slow tail such that the equilibrium gain is not recovered in a cavity round-trip time. This ultra-slow gain recovery or non-recovered gain cannot be explained by only the intersubband carrier dynamics of QCLs. This work shows that the Fabry-Perot cavity dynamics and localized intersubband electron heating of QCLs are essential in ultra-slow and nonrecovered gain recovery. We developed a comprehensive model, coupling cavity dynamics to the intersubband electrons' thermal evolution. We employ a four-level coupled Maxwell-Bloch model that considers temperature-dependent scattering and transport mechanisms in calculating the gain recovery dynamics. If an intense pump pulse electrically pumped close to the threshold propagates in the forward direction after being coupled into the cavity, the reflected pump pulse will significantly deplete the gain medium while propagating in the backward direction. Additionally, we show that the intersubband electron sustains a localized high temperature even after the pump pulse has left, which affects the overall carrier dynamics and leads to an ultra-slow gain recovery process. At near-perfect reflectivity, we observe a gain depletion of 4% for 2 mm QCL. We further demonstrate that an additional 10% gain depletion of probe pulse is seen at a steady state when the laser is pumped at 1.6 times the threshold compared to the case where the hot electron effect is not considered