  We review recent investigations of the femtosecond non-linear optical
response of the two-dimensional electron gas (2DEG) in a strong magnetic field.
We probe the Quantum Hall (QH) regime for filling factors $\nu \sim 1$. Our
focus is on the transient coherence induced via optical excitation and on its
time evolution during early femtosecond timescales. We simultaneously study the
interband and intraband coherence in this system by using a nonlinear
spectroscopic technique, transient three-pulse four wave mixing optical
spectroscopy, and a many-body theory. We observe striking differences in the
temporal and spectral profile of the nonlinear optical signal between a
modulation doped quantum well system (with the 2DEG) and a similar undoped
quantum well (without a 2DEG). We attribute these qualitative differences to
Coulomb correlations between the photoexcited electron-hole pairs and the 2DEG.
We show, in particular, that intraband many-particle coherences assisted by the
inter-Landau-level magnetoplasmon excitations of the 2DEG dominate the
femtosecond nonlinear optical responce. The most striking effect of these
exciton-magnetoplasmon coherences is a large off-resonant four-wave-mixing
signal in the case of very low photoexcited carrier densities, not observed in
the undoped system, with strong temporal oscillations and unusually symmetric
temporal profile.
