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We carry out a study on the validity and limitations of truncation schemescustomarily employed to treat the quantum kinetic equations of motion ofcomplex interacting systems. Our system of choice is a semiconductor quantumring with one electron interacting with few magnetic impurities via aKondo-like Hamiltonian. This system is an interesting prototype which displaysthe necessary complexity when suitably scaled (large number of magneticimpurities) but can also be solved exactly when few impurities are present. Thecomplexity in this system comes from the indirect electron-mediatedimpurity-impurity interaction and is reflected in the Heisenberg equations ofmotion, which form an infinite hierarchy. For the cases of two and threemagnetic impurities, we solve for the quantum dynamics of our system bothexactly and following a truncation scheme developed for diluted magneticsemiconductors in the bulk. We find an excellent agreement between the twoapproaches when physical observables like the impurities' spin angular momentumare computed for times that well exceed the time window of validity ofperturbation theory. On the other hand, we find that within time ranges ofphysical interest, the truncation scheme introduces negative populations whichrepresents a serious methodological drawback.

Case study of the validity of truncation schemes of kinetic equations of motion: few magnetic impurities in a semiconductor quantum ring