  Coherent light-matter interaction can be used to manipulate the energy levels
of atoms, molecules and solids. When light with frequency {\omega} is detuned
away from a resonance {\omega}o, repulsion between the photon-dressed (Floquet)
states can lead to a shift of energy resonance. The dominant effect is the
optical Stark shift (1/({\omega}0-{\omega})), but there is an additional
contribution from the so-called Bloch-Siegert shift (1/({\omega}o+{\omega})).
Although it is common in atoms and molecules, the observation of Bloch-Siegert
shift in solids has so far been limited only to artificial atoms since the
shifts were small (<1 {\mu}eV) and inseparable from the optical Stark shift.
Here we observe an exceptionally large Bloch-Siegert shift (~10 meV) in
monolayer WS2 under infrared optical driving by virtue of the strong
light-matter interaction in this system. Moreover, we can disentangle the
Bloch-Siegert shift entirely from the optical Stark shift, because the two
effects are found to obey opposite selection rules at different valleys. By
controlling the light helicity, we can confine the Bloch-Siegert shift to occur
only at one valley, and the optical Stark shift at the other valley. Such a
valley-exclusive Bloch-Siegert shift allows for enhanced control over the
valleytronic properties in two-dimensional materials, and offers a new avenue
to explore quantum optics in solids.
