  Coherent optical manipulation of electronic bandstructures via Floquet
Engineering is a promising means to control quantum systems on an ultrafast
timescale. However, the ultrafast switching on/off of the driving field comes
with questions regarding the limits of validity of the Floquet formalism, which
is defined for an infinite periodic drive, and to what extent the transient
changes can be driven adibatically. Experimentally addressing these questions
has been difficult, in large part due to the absence of an established
technique to measure coherent dynamics through the duration of the pulse. Here,
using multidimensional coherent spectroscopy we explicitly excite, control, and
probe a coherent superposition of excitons in the $K$ and $K^\prime$ valleys in
monolayer WS$_2$. With a circularly polarized, red-detuned, pump pulse, the
degeneracy of the $K$ and $K^\prime$ excitons can be lifted and the phase of
the coherence rotated. We demonstrate phase rotations during the 100 fs driving
pulse that exceed $\pi$, and show that this can be described by a combination
of the AC-Stark shift of excitons in one valley and Bloch-Siegert shift of
excitons in the opposite valley. Despite showing a smooth evolution of the
phase that directly follows the intensity envelope of the pump pulse, the
process is not perfectly adiabatic. By measuring the magnitude of the
macroscopic coherence as it evolves before, during, and after the pump pulse we
show that there is additional decoherence caused by power broadening in the
presence of the pump. This non-adiabaticity may be a problem for many
applications, such as manipulating q-bits in quantum information processing,
however these measurements also suggest ways such effects can be minimised or
eliminated.
