Quantum dynamics of two quantum dots in the vicinity of a photonic band edge: role of structured reservoirs and phonons

We study the population and disentanglement dynamics of two identical quantum dots placed inside the structured electromagnetic reservoir of a photonic crystal and coupled to an independent bath of thermal phonons. A formalism based on the method of generalized-Laplace transforms is developed to study the population and disentanglement dynamics. The effect of resonant dipole dipole interaction between the two quantum dots on the population dynamics in the presence of phonons and shows that the two two-level atoms can have a residual population at long times in the structured reservoir of a photonic crystal due to the formation of a photon-atom bound state. The disentanglement dynamics of the two quantum dots (assumed to be initially entangled) is studied using concurrence as a measure and is computed from the reduced two two-level system density matrix using the method of generalized-Laplace transforms. We show that partial disentanglement results from the interaction of the two quantum dots with the electromagnetic and phonon reservoirs. However, substantial entanglement is preserved in the fractionalized steady state formed if the transition frequency of two the quantum dots lies in the vicinity of the photonic band edge. We also discuss in detail the role of acoustic phonons on the population and disentanglement dynamics and the long time residual entanglement.