  We present and analyze a method where parametric (two-photon) driving of a
cavity is used to exponentially enhance the light-matter coupling in a generic
cavity QED setup, with time-dependent control. Our method allows one to enhance
weak-coupling systems, such that they enter the strong coupling regime (where
the coupling exceeds dissipative rates) and even the ultra-strong coupling
regime (where the coupling is comparable to the cavity frequency). As an
example, we show how the scheme allows one to use a weak-coupling system to
adiabatically prepare the highly entangled ground state of the ultra-strong
coupling system. The resulting state could be used for remote entanglement
applications.
