  Light-matter coupling involving classical and quantum light offers a wide
range of possibilities to tune the electronic properties of correlated quantum
materials. Two paradigmatic results are the dynamical localization of electrons
and the ultrafast control of spin dynamics, which have been discussed within
classical Floquet engineering and in the deep quantum regime where vacuum
fluctuations modify the properties of materials. Here we discuss how these two
extreme limits are interpolated by a cavity which is driven to the excited
states. In particular, this is achieved by formulating a Schrieffer-Wolff
transformation for the cavity-coupled system, which is mathematically analogous
to its Floquet counterpart. Some of the extraordinary results of
Floquet-engineering, such as the sign reversal of the exchange interaction or
electronic tunneling, which are not obtained by coupling to a dark cavity, can
already be realized with a single-photon state (no coherent states are needed).
The analytic results are verified and extended with numerical simulations on a
two-site Hubbard model coupled to a driven cavity mode. Our results generalize
the well-established Floquet-engineering of correlated electrons to the regime
of quantum light. It opens up a new pathway of controlling properties of
quantum materials with high tunability and low energy dissipation.
