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We investigate coupling of localized spins in a semiconductor quantum dotembedded in a microcavity with a finite quality factor. The lowest cavity modeand the quantum dot exciton are coupled forming a polariton, whereas excitonsinteract with localized spins via exchange. The finite quality of the cavity Qis incorporated in the model Hamiltonian by adding an imaginary part to thephoton frequency. The Hamiltonian, which treats photons, spins and excitonsquantum mechanically, is solved exactly. Results for a single polariton clearlydemonstrate the existence of a resonance, sharper as the temperature decreases,that shows up as an abrupt change between ferromagnetic and antiferromagneticindirect anisotropic exchange interaction between localized spins. The originof this spin-switching finite-quality-factor effect is discussed in detailremarking on its dependence on model parameters, i.e., light-matter coupling,exchange interaction between impurities, detuning and quality factor. Forparameters corresponding to the case of a (Cd,Mn)Te quantum dot, the resonanceshows 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 (qualityfactors better than 200) the exchange interaction is scarcely affected.Comment: 7 figures, submitted to PR

Switching the sign of photon-induced exchange interactions in semiconductor microcavities with finite quality factors