  High-order topological phases, such as those with nontrivial quadrupole
moments, protect edge states that are themselves topological insulators in
lower dimensions. So far, most quadrupole phases of light are explored in
linear optical systems, which are protected by spatial symmetries or synthetic
symmetries. Here we present Floquet quadrupole phases in driven nonlinear
photonic crystals (PhCs) that are protected by space-time screw symmetries. We
start by illustrating space-time symmetries by tracking the trajectory of
instantaneous optical axes of the driven media. Our Floquet quadrupole phase is
then confirmed in two independent ways: symmetry indices at high-symmetry
momentum points and calculations of the nested Wannier bands. Our work presents
a general framework to analyze symmetries in driven optical materials and paves
the way to further exploring symmetry-protected topological phases in Floquet
systems and their optoelectronic applications.
