Floquet engineering in the presence of optically excited carriers

Floquet engineering provides an optical means to manipulate electronic bandstructures, however, carriers excited by the pump field can lead to an effective heating, and can obscure measurement of the band changes. A recent demonstration of the effects of Floquet engineering on a coherent ensemble of excitons in monolayer WS$_2$ proved particularly sensitive to non-adiabatic effects, while still being able to accurately resolve bandstructure changes. Here, we drive an AC-Stark effect in monolayer WS$_2$ using pulses with constant fluence but varying pulse duration (from 25-235~fs). With shorter pump pulses, the corresponding increase in peak intensity introduces additional carriers via two-photon absorption, leading to additional decoherence and peak broadening (which makes it difficult to resolve the AC-Stark shift). We use multidimensional coherent spectroscopy to create a coherent ensemble of excitons in monolayer WS$_2$ and measure the evolution of the coherence throughout the duration of the Floquet pump pulse. Changes to the amplitude of the macroscopic coherence quantifies the additional broadening. At the same time, the evolution of the average phase allows the instantaneous changes to the bandstructure to be quantified, and is not impacted by the additional broadening. This approach to measuring the evolution of Floquet-Bloch states demonstrates a means to quantify effective heating and non-adiabaticity caused by excited carriers, while at the same time resolving the coherent evolution of the bandstructure.