We analyze the dynamics of spin-mixing interactions generated by coupling spin-1 atoms to the mode of a high-finesse optical cavity. We show that the dynamics can be understood in terms of generators of the noncompact 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. Non-uniform coupling between the atomic ensemble and the cavity mode leads to departures from this simple behavior; we introduce a toy model that captures this non-uniformity and solve it exactly.

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Gregory S. Bentsen, Emily J. Davis, Lukas Homeier, Avikar Periwal, Eric Cooper, Katherine Van Kirk, and Monika H. Schleier-Smith, "Photon-mediated spin-mixing dynamics," Proc. SPIE 10934, Optical, Opto-Atomic, and Entanglement-Enhanced Precision Metrology, 109342P (Presented at SPIE OPTO: February 07, 2019; Published: 1 March 2019); https://doi.org/10.1117/12.2515795.