Dynamics of a harmonic oscillator in a large-S magnetic environment

We consider a composite system made of a quantum harmonic oscillator and a magnetic system with fixed total spin S, interacting via a Dicke-like Hamiltonian. In the large-S limit, the global unitary dynamics is found to result from the composition of two propagators, separately describing the energy exchange from the oscillator to the environment, and viceversa. The corresponding reduced dynamics for the oscillator is then studied by the parametric representation with environmental coherent states, and the case when the environment is initially prepared in the state with Sz=-S is specifically addressed. Connections with the behaviour of an oscillator in a fluctuating classical field emerge, as well as indications about the role played by the internal symmetry of the environment, that ensures S is constant.