Crossing Phantom Boundary in $f(R)$ Modified Gravity : Jordan Frame vs Einstein Frame

We study capability of $f(R)$ gravity models to allow crossing the phantom boundary in both Jordan and Einstein conformal frames. In Einstein frame, these models are equivalent to Einstein gravity together with a scalar field minimally coupled to gravity. This scalar degree of freedom appears as a quintessence field with a coupling with the matter sector. We investigate evolution of the equation of sate parameter for some cosmologically viable $f(R)$ gravity models in both conformal frames. This investigation (beyond mere theoretical arguments) acts as an operational tool to distinguish physical status of the two conformal frames. It shows that the two conformal frames have not the same physical status.