Photon-mediated spin-mixing dynamics

We analyze the dynamics of spin-mixing interactions generated by coupling spin-1 atoms to the mode of a high-fi nesse optical cavity. We show that the dynamics can be understood in terms of generators of the non-compact Lie group SU(1; 1) and introduce a set of SU(1; 1) coherent states which are preserved under Hamiltonian evolution. In terms of these coherent states the resulting dynamics may be interpreted as classical motion on the unit disk. We explicitly compute the trajectories of this classical motion and show that the motion is equivalent to spin-nematic squeezing in the atomic ensemble. Dissipation due to photon loss and non-uniform coupling between the atomic ensemble and the cavity mode lead to departures from this simple behavior. We introduce toy models that capture these experimental imperfections and solve them exactly.