Dynamics of a periodic $XY$ chain coupled to a photon mode

We study the real-time dynamics of a periodic $XY$ system exposed to a composite field comprised of a constant homogeneous magnetic and a quantized circularly polarized electromagnetic fields. The interaction between the quantized mode and spin-magnetic moments is modeled by the Dicke Hamiltonian. The rotating wave approximation is applied and the conditions for its validity are discussed. It is shown that if at the beginning of the dynamic process all of the excitations are contained in the field then, in the regime of large detuning, the main evolutionary effect is the oscillations of the excitations between the zero-momentum modes of the chain and the field. Accordingly, the reduced photon number and magnetization per site reveal a sort of oscillatory behavior. Effective Hamiltonians approximating the short-time dynamics of the actual problem for small number of excitations and large detuning are introduced. The resonance case is considered in the context of photon emission from the chain initially prepared in the (partially) excited state. In particular, it is demonstrated in the framework of a specific example that the superradiant behavior can be presented in the beginning of the emission, when the evaluation starts from maximally excited $XY$ chain. The model is potentially applicable to problems such as spin chain/linear molecular aggregates in a single-mode cavity.