Dissipation and resonance frequency shift of a resonator magnetically coupled to a semiclassical spin

We calculate the change of the properties of a resonator, when coupled to a semiclassical spin by means of the magnetic field. Starting with the Lagrangian of the complete system, we provide an analytical expression for the linear response function for the motion in the case of a mechanical resonator and the current for the case of an electromagnetic resonator, thereby considering the influence of the resonator on the spin and vice versa. This analysis shows that the resonance frequency and effective dissipation factor can change significantly due to the relaxation times of the spin. We first derive this for a system consisting of a spin and mechanical resonator and thereafter apply the same calculations to an electromagnetic resonator. Moreover, the applicability of the method is generalized to a resonator coupled to two-level systems and more, providing a key to understand some of the problems of two-level systems in quantum devices.