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We study the spin orbit coupled ultra cold Bose-Einstein condensate placed ina single mode Fabry-P\'erot cavity. The cavity introduces a quantum opticallattice potential which dynamically couples with the atomic degrees of freedomand realizes a generalized extended Bose Hubbard model whose zero temperaturephase diagram can be controlled by tuning the cavity parameters. In thenon-interacting limit, where the atom-atom interaction is set to zero, theresulting atomic dispersion shows interesting features such as bosonic analogueof Dirac points, cavity controlled Hofstadter spectrum which bears the hallmarkof 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 latticepotential. In the presence of atom-atom interaction, using a mapping to ageneralized Bose Hubbard model of spin-orbit coupled bosons in classicaloptical lattice, we show that the system realizes a host of quantum magneticphases whose magnetic order can be be detected from the cavity transmission.This provides an alternative approach for detecting quantum magnetism in ultracold atoms. We discuss the effect of cavity induced optical bistability on thisphases and their experimental consequences.

Ultracold Spin-Orbit Coupled Bose-Einstein Condensate in a Cavity: Route to Magnetic Phases Through Cavity Transmission