Effective Hamiltonian for the impurity-impurity interaction in micropillars in the presence of decoherence

We study the optically induced effective interaction between spins in micropillars. Our theoretical model describes a circularly polarized cavity mode strongly coupled to two quantum-dot excitons, each of which interacts with a localized spin. The decoherence effect inducing a finite lifetime to the cavity mode is included. Using the master equation for the density matrix we investigate the dynamics under low-power excitation. In the absence of decoherence, we recover the result obtained in a previous work (which uses conventional low-energy reduction procedures) for an effective Hamiltonian containing a Zeeman term and an Ising interaction between the localized spins. For finite cavity-photon lifetime we employ Fourier analysis on the time dependence of the nondiagonal element of the reduced density matrix, obtained by tracing out the photons and excitons, to find a quasieffective spin-spin Ising Hamiltonian. This Hamiltonian is a possible building block for two-qubit quantum computing operation. We study how decoherence affects the construction of that effective interaction. Finally, we discuss the possible application for transition-metal impurity spins embedded in CdTe quantum dots.