  We study Einstein-Podolsky-Rosen (EPR) steering properties of quantum phases
in the generalized Dicke model (GDM) generated by a cavity Bose-Einstein
condensate (BEC) doped with a single impurity. It is shown that the normal and
superradiant phases of the GDM exhibit much rich EPR steerability. In the
normal phase, there exist one-way EPR steering from the cavity field (condensed
atoms) to condensed atoms (the cavity field). In the superradiant phase, there
is either one-way or two-way EPR steering for the cavity field and condensed
atoms. It is found that the single impurity can act as a single-atom switch of
the EPR steering to control the steering direction of one-way EPR steering and
the transition between one-way and two-way EPR steering through switching on or
off the impurity-BEC coupling. It is proved that the EPR steering parameters
can witness the quantum phase transition (QPT) occurrence through the sudden
change of the EPR steering parameters at the critical point of the QPT. Our
results open a new way to manipulate macroscopic EPR steering and witness the
QPT between the normal phase and the superradiant phase in the GDM through a
microscopic impurity atom.
