  We consider a new approach to the problem of Bose-Einstein condensation (BEC)
of polaritons for atom-field interaction under the strong coupling regime in
the cavity. We investigate the dynamics of two macroscopically populated
polariton modes corresponding to the upper and lower branch energy states
coupled via Kerr-like nonlinearity of atomic medium. We found out the
dispersion relations for new type of collective excitations in the system under
consideration. Various temporal regimes like linear (nonlinear) Josephson
transition and/or Rabi oscillations, macroscopic quantum self-trapping (MQST)
dynamics for population imbalance of polariton modes are predicted. We also
examine the switching properties for time-averaged population imbalance
depending on initial conditions, effective nonlinear parameter of atomic medium
and kinetic energy of low-branch polaritons.
