  The theory of phase control of coherence, entanglement and quantum steering
is developed for an optomechanical system composed of a single mode cavity
containing a partially transmitting dielectric membrane and driven by short
laser pulses. The closed loop in the coupling creates interfering channels
which depend on the relative phase of the coupling strengths of the field modes
to the mechanical mode. We show several interesting phase dependent effects
such as reversible population transfer from one field mode to the other,
creation of collective modes, and induced coherence without induced emission.
These effects result from perfect mutual coherence between the field modes
which is preserved even if one of the modes is not populated. Depending on the
phase, the field modes can act on the mechanical mode collectively or
individually resulting, respectively, in tripartite or bipartite entanglement.
In addition, we examine the phase sensitivity of quantum steering of the
mechanical mode by the field modes is investigated. Deterministic phase
transfer of the steering from bipartite to collective is predicted and optimum
steering corresponding to perfect EPR state can be achieved. These different
types of quantum steering can be distinguished experimentally by measuring the
coincidence rate between two detectors adjusted to collect photons of the
output cavity modes. In particular, we find that the minima of the interference
pattern of the coincidence rate signal the bipartite steering, while the maxima
signal the collective steering.
