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Coherent optical manipulation of electronic bandstructures via FloquetEngineering is a promising means to control quantum systems on an ultrafasttimescale. However, the ultrafast switching on/off of the driving field comeswith questions regarding the limits of validity of the Floquet formalism, whichis defined for an infinite periodic drive, and to what extent the transientchanges can be driven adibatically. Experimentally addressing these questionshas been difficult, in large part due to the absence of an establishedtechnique to measure coherent dynamics through the duration of the pulse. Here,using multidimensional coherent spectroscopy we explicitly excite, control, andprobe a coherent superposition of excitons in the $K$ and $K^\prime$ valleys inmonolayer WS$_2$. With a circularly polarized, red-detuned, pump pulse, thedegeneracy of the $K$ and $K^\prime$ excitons can be lifted and the phase ofthe coherence rotated. We demonstrate phase rotations during the 100 fs drivingpulse that exceed $\pi$, and show that this can be described by a combinationof the AC-Stark shift of excitons in one valley and Bloch-Siegert shift ofexcitons in the opposite valley. Despite showing a smooth evolution of thephase that directly follows the intensity envelope of the pump pulse, theprocess is not perfectly adiabatic. By measuring the magnitude of themacroscopic coherence as it evolves before, during, and after the pump pulse weshow that there is additional decoherence caused by power broadening in thepresence of the pump. This non-adiabaticity may be a problem for manyapplications, such as manipulating q-bits in quantum information processing,however these measurements also suggest ways such effects can be minimised oreliminated.

Effects of Floquet Engineering on the Coherent Exciton Dynamics in Monolayer WS$_2$