  By using the Kubo linear response theory with the Keldysh Green function
approach, we investigate the mechanism leading to the negative differential
transmission in system with the equilibrium electron density much smaller than
the photon-excited one. It is shown that the negative differential transmission
can appear at low probe-photon energy (in the order of the scattering rate) or
at high energy (much larger than the scattering rate). For the low probe-photon
energy case, the negative differential transmission is found to come from the
increase of the intra-band conductivity due to the large variation of electron
distribution after the pumping. As for the high probe-photon energy case, the
negative differential transmission is shown to tend to appear with the
hot-electron temperature being closer to the equilibrium one and the chemical
potential higher than the equilibrium one but considerably smaller than half of
the probe-photon energy. We also show that this negative differential
transmission can come from both the inter- and the intra-band components of the
conductivity. Especially, for the inter-band component, its contribution to the
negative differential transmission is shown to come from both the Hartree-Fock
self-energy and the scattering. Furthermore, the influence of the Coulomb-hole
self-energy is also addressed.
