Radiative coupling of quantum dots in a disordered Photonic Crystal waveguide

Abstract

Summary form only given. Semiconductor quantum dots (QDs) have very recently become candidate building blocks of a quantum information technology, after the experimental proof of full single-qubit control. An essential requirement of the quantum information paradigm is the possibility for two qubits to interact coherently in a controlled fashion. Given the recent success in embedding QDs in Photonic Crystal (PHC) devices, photons could provide such a quantum bus, mediating the coherent excitation exchange between two distant dots. Here, we present a comprehensive theoretical analysis of the magnitude and distance dependence of the radiative coupling of two dots in a PHC W1 waveguide, based on Maxwell equations and on the linear excitonic response of the QDs [1].The coupling is quantified through the Green's function Gαβ(ω0), evaluated at the dot positions rα and rβ, and the exciton transition resonance frequency ω0. Here, Gαβ is determined by the microscopic properties of both the quantum dots and the photonic structure (relevant for state-of-the-art systems). Fabrication disorder is introduced in the form of Gaussian fluctuations with a standard deviation σ of the PHC hole positions and radii. The PHC modes can be efficiently and reliably computed using an expansion on the Bloch modes of the regular waveguide. A waveguide of length 512a (a = 260 nm) is simulated, and |Gαβ(ω0)| is averaged over several hundred disorder realizations and all possible QD positions. The dependence of this quantity on inter-dot distance is well described by an exponential law, |G12|≈ |G11|exp(-x/L), where G11 is the self-energy of the first dot (which determines the Purcell factor), x = r2 - r1, and L is a characteristic attenuation length. In Fig. 1 (a), (b), we show the dependence of |G11| and L on ω0 for frequencies close to the guided band edge.In a disorder-less waveguide, the modes close to the band edge typically lie mostly outside the light cone and are thus loss-less, allowing excitation transfer between the QDs at arbitrarily large distances. Disorder has two effects: first, it scatters light above the light cone, thus introducing extrinsic out-of-plane losses, and second, it introduces Anderson localization of light (Fig 1. (c)). With only extrinsic losses taken into account, the characteristic length for the interaction is shown [1] to be L= 2vg/γ, with vg the group velocity at ω0, and γ = ω0/Q the average extrinsic loss rate. In Fig. 1 (b), this quantity is plotted with thin lines for Q = 15 000 (average value for σ = 0.01a) and Q = 95 000 (average value for σ = 0.004a). Field localization changes this length drastically, as evidenced by the thick lines in the figure, for which the attenuation length L was extracted by numerical interpolation of the distance decay of |G12(ω0)|. Our results suggest that for realistic disorder, the dot-dot coupling rate is larger than 10 μeV at distances of the order of a hundred unit cells, thus longer than 20μm. We show how this value fluctuates greatly with the particular disorder realization, frequency ω0, and dot positions - and can be much larger in specific cases. Our results clearly show that, in a realistic system with realistic disorder, the exchange time is of the order of 10 ps, namely close to the single-qubit operation time and much shorter than the decoherence time measured in these systems. This suggests that a PHC waveguide - or a similar structure engineered to further enhance the dot-dot coupling - is an ideal candidate for establishing a long-distance, photon-mediated QD state transfer.