We use the terahertz (THz) emission spectroscopy to study femtosecond photocurrent dynamics in the prototypical 2D semiconductor, transition metal dichalcogenide MoSe$_2$. We identify several distinct mechanisms producing THz radiation in response to an ultrashort ($30,$fs) optical excitation in a bilayer (BL) and a multilayer (ML) sample. In the ML, the THz radiation is generated at a picosecond timescale by out-of-plane currents due to the drift of photoexcited charge carriers in the surface electric field. The BL emission is generated by an in-plane shift current. Finally, we observe oscillations at about $23,$THz in the emission from the BL sample. We attribute the oscillations to quantum beats between two excitonic states with energetic separation of $sim100,$meV.
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