  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.
