  Inducing novel quantum phases and topologies in materials using intense light
fields is a key objective of modern condensed matter physics, but nonetheless
faces significant experimental challenges. Alternately, theory predicts that in
the dense limit, excitons - collective excitations composed of Coulomb-bound
electron-hole pairs - could also drive exotic quantum phenomena. However, the
direct observation of these phenomena requires the resolution of electronic
structure in momentum space in the presence of excitons, which became possible
only recently. Here, using time- and angle-resolved photoemission spectroscopy
of an atomically thin semiconductor in the presence of a high-density of
resonantly and coherently photoexcited excitons, we observe the
Bardeen-Cooper-Schrieffer (BCS) excitonic state - analogous to the Cooper pairs
of superconductivity. We see the valence band transform from a conventional
paraboloid into a Mexican-hat like Bogoliubov dispersion - a hallmark of the
excitonic insulator phase; and we observe the recently predicted giant
exciton-driven Floquet effects. Our work realizes the promise that intense
bosonic fields, other than photons, can also drive novel quantum phenomena and
phases in materials.
