  We study the spin orbit coupled ultra cold Bose-Einstein condensate placed in
a single mode Fabry-P\'erot cavity. The cavity introduces a quantum optical
lattice potential which dynamically couples with the atomic degrees of freedom
and realizes a generalized extended Bose Hubbard model whose zero temperature
phase diagram can be controlled by tuning the cavity parameters. In the
non-interacting limit, where the atom-atom interaction is set to zero, the
resulting atomic dispersion shows interesting features such as bosonic analogue
of Dirac points, cavity controlled Hofstadter spectrum which bears the hallmark
of pseudo-spin-1/2 bosons in presence of Abelian and non-Abelian gauge field
(the later due to spin-orbit coupling) in a cavity induced optical lattice
potential. In the presence of atom-atom interaction, using a mapping to a
generalized Bose Hubbard model of spin-orbit coupled bosons in classical
optical lattice, we show that the system realizes a host of quantum magnetic
phases whose magnetic order can be be detected from the cavity transmission.
This provides an alternative approach for detecting quantum magnetism in ultra
cold atoms. We discuss the effect of cavity induced optical bistability on this
phases and their experimental consequences.
