Momchil Minkov, Vincenzo Savona
We derive a general formalism to model the polariton states resulting from the radiation-matter interaction between an arbitrary number of excitonic transitions in semiconductor quantum dots and photon modes in a photonic crystal structure in which the quantum dots are embedded. The Maxwell equations, including the linear nonlocal susceptibility of the exciton transitions in the quantum dots, are cast into an eigenvalue problem, where the matrix elements are expressed as functions of the microscopic parameters of the semiconductor system. We compute realistic photon modes using Bloch-mode expansion, while the simplest assumption of a delta-like excitonic wave-function for the quantum dots is made. As examples, we look at one or two quantum dots embedded in a photonic crystal with a line defect -- an $Ln$ cavity or a $\mathit{W1}$ waveguide. For the single dot case, we reproduce known analytical results, while for the two dot case, we study the radiative excitation transfer mechanism and characterize its strength, the dependence on the detuning between quantum dot and photon modes, and the dependence on inter-dot distance. We find in particular that, for typical InGaAs quantum dots in a GaAs photonic crystal structure, the inter-dot radiative coupling strength can reach $100 \mu{eV}$ in a short cavity, and its decay with distance in longer cavities and waveguides is determined by the group velocity of the exchanged photons and their radiative lifetime. We finally show that, for an $Ln$ cavity of increasing length, the radiative excitation transfer mechanism is subject to a crossover from a regime where a single photon mode is dominating, to a multi-mode regime. The developed numerical procedure can in principle be applied to any structure whose photon modes can be computed with reliable accuracy, and in addition naturally allows for disorder effects to be taken into account.
View original:
http://arxiv.org/abs/1212.4960
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