  Subjecting a physical system to a time-periodic drive can substantially
modify its properties and applications. This Floquet-engineering approach has
been extensively applied to a wide range of classical and quantum settings in
view of designing synthetic systems with exotic properties. Considering a
general class of two-mode nonlinear optical devices, we show that effective
optical nonlinearities can be created by subjecting the light field to a
repeated pulse sequence, which couples the two modes in a fast and
time-periodic manner. The strength of these drive-induced optical
nonlinearities, which include an emerging four-wave mixing, can be varied by
simply adjusting the pulse sequence. This leads to topological changes in the
system's phase space, which can be detected through light intensity and phase
measurements. Our proposal builds on an effective-Hamiltonian approach, which
derives from a parent quantum many-body Hamiltonian describing driven
interacting bosons. As a corollary, our results equally apply to Bose-Einstein
condensates in driven double-well potentials, where pair tunneling effectively
arises from the periodic pulse sequence. Our scheme offers a practical route to
engineer and finely tune exotic nonlinearities and interactions in photonics
and ultracold quantum gases.
