  We theoretically study the coherent nonlinear response of electrons confined
in semiconductor quantum wells under the effect of an electromagnetic radiation
close to resonance with an intersubband transition. Our approach is based on
the time-dependent Schr\"odinger-Poisson equation stemming from a Hartree
description of Coulomb-interacting electrons. This equation is solved by
standard numerical tools and the results are interpreted in terms of
approximated analytical formulas. For growing intensity, we observe a redshift
of the effective resonance frequency due to the reduction of the electric
dipole moment and the corresponding suppression of the depolarization shift.
The competition between coherent nonlinearities and incoherent saturation
effects is discussed. The strength of the resulting optical nonlinearity is
estimated across different frequency ranges from mid-IR to THz with an eye to
ongoing experiments on Bose-Einstein condensation of intersubband polaritons
and to the speculative exploration of quantum optical phenomena such as
single-photon emission in the mid-IR and THz windows.
