  Low electron density Drude (LEDD) materials such as indium tin oxide (ITO)
are receiving considerable attention because of their combination of CMOS
compatibility, unique epsilon-near-zero (ENZ) behavior, and giant ultrafast
nonlinear thermo-optic response. However, the understanding of the electronic
and optical response of LEDD materials is so far based on simplistic extensions
of known models of noble metals, frequently without the inclusion of the
interplay among the lower electron density, relatively high Debye energy, and
the non-parabolic band structure. To bridge this knowledge gap, this work
provides a complete understanding of the nonlinear electronic-thermal-optical
response of LEDD materials. In particular, we rely on state-of-the-art electron
dynamics modeling, as well as the newly derived time-dependent permittivity
model for LEDD materials under optical pumping within the adiabatic
approximation. We find that unlike noble metals, the electron temperatures can
reach the Fermi temperature, in which case the effective chemical potential
dramatically decreases and even becomes negative, thus, transiently converting
the Drude metal into a semiconductor. We further show that the nonlinear
optical response of LEDD materials originating from the changes to the real
part of the permittivity is due to the generation of non-thermal electrons.
This resolves the argument about the rise time of the permittivity and shows
that it is instantaneous. In this vein, we show that referring to the LEDD
permittivity as having a ``saturable'' nonlinearity is unsuitable since its
permittivity dynamics does not originate from population inversion. Finally, we
analyze the probe pulse dynamics and unlike previous work, we obtain a
quantitative agreement with the results of recent experiments.
