  We investigate coupling of localized spins in a semiconductor quantum dot
embedded in a microcavity with a finite quality factor. The lowest cavity mode
and the quantum dot exciton are coupled forming a polariton, whereas excitons
interact with localized spins via exchange. The finite quality of the cavity Q
is incorporated in the model Hamiltonian by adding an imaginary part to the
photon frequency. The Hamiltonian, which treats photons, spins and excitons
quantum mechanically, is solved exactly. Results for a single polariton clearly
demonstrate the existence of a resonance, sharper as the temperature decreases,
that shows up as an abrupt change between ferromagnetic and antiferromagnetic
indirect anisotropic exchange interaction between localized spins. The origin
of this spin-switching finite-quality-factor effect is discussed in detail
remarking on its dependence on model parameters, i.e., light-matter coupling,
exchange interaction between impurities, detuning and quality factor. For
parameters corresponding to the case of a (Cd,Mn)Te quantum dot, the resonance
shows up for Q around 70 and detuning around 10 meV. In addition, we show that,
for such a quantum dot, and the best cavities actually available (quality
factors better than 200) the exchange interaction is scarcely affected.
