Cavity-mediated dissipative spin-spin coupling

We study dissipative spin-spin coupling in a dispersive regime mediated by virtual photons in a microwave cavity. Dissipative coupling between spin of each magnetic material and the cavity photons is established by means of two phase-shifted driving forces acting on each spin. We show that mode-level attraction between two spin modes can be reached when one of the spins is dissipatively coupled to the cavity. By tuning the phase parameter at each ferromagnetic insulator, we can shift the order of "dark" and "bright" collective modes with the phase difference equal to 0 or pi. Moreover, by selectively applying the phase-shifted field, it is possible to construct dark and bright collective modes with the phase difference equal to +/-pi/2.