Generalized Holstein-Primakoff mapping and $1/N$ expansion of collective spin systems undergoing single particle dissipation

We develop a generalization of the Schwinger boson and Holstein-Primakoff transformations that is applicable to ensembles of N spin $1/2$'s with weak permutational symmetry. These generalized mappings are constructed by introducing two independent bosonic variables that describe fluctuations parallel and transverse to the collective Bloch vector built out of the original spin $1/2$'s. Using this representation, we develop a systematic $1/N$ expansion and write down explicitly leading and next-to-leading order terms. We then illustrate how to apply these techniques using four example systems: (i) an ensemble of atoms undergoing spontaneous emission, incoherent pumping and single particle dephasing; (ii) a superradiant laser above and in the vicinity of the upper lasing transition; (iii) the all-to-all transverse field Ising model subject to incoherent pumping in the vicinity of its ordering phase transition; and (iv) the Dicke model at finite temperature both away and in the vicinity of its thermal phase transition. Thus, these mappings provide a common, Bloch-sphere based, geometrical description of all-to-all systems subject to single particle dissipation or at finite temperature, including their phase transitions.