Dimensional crossover in self-organised super-radiant lattice supersolid phases of ultra cold atoms inside a cavity

We consider a condensate of ultra cold bosonic atoms in a linear optical cavity illuminated by a two-pump configuration where each pump is making different angles with the direction of the cavity axis. We show such configuration allows a smooth transition from a one-dimensional quantum optical lattice configuration to a two-dimensional quantum optical lattice configuration induced by the cavity-atom interaction. Using a Holstein-Primakoff transformation, we find out the atomic density profile of such self-organised ground state in the super-radiant phase as a function of the angular orientations of the pump in such dynamical quantum optical lattice . We also compare our results with the one calculated using Extended Bose-Hubbard model in such quantum optical lattice potential.